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3opy 2 


An Experimental Study 

TO DETERMINE THE 

RELATIVE DIFFICULTY OF THE ELEMENTARY 
NUMBER COMBINATIONS IN ADDITION 
AND MULTIPLICATION. 


“A thesis presented to the Faculty of the Graduate School of 
the University of Pennsylvania in partial fulfilment of the 
requirements for the degree of Doctor of Philosophy,” 

BY 

HARRY VANCE HOLLOWAY, 

June, 1914, 



TRENTON, N. J. 

State Gazette Publishing Co., Printers, 
1915. 








An Experimental Study 

TO DETERMINE THE 

RELATIVE DIFFICULTY OF THE ELEMENTARY 
NUMBER COMBINATIONS IN ADDITION 
AND MULTIPLICATION. 


“A thesis presented to the Faculty of the Graduate School of 
the University of Pennsylvania in partial fulfilment of the 
requirements for the degree of Doctor of Philosophy.” 


BY 


HARRY VANCE HOLLOWAY. 


June, 1914. 



TRENTON, N. J. 

State Gazette Publishing Co., Printers. 
1915 . 






* 




















Contents 


) 


f 


) 

€ 


I. The need of such an investigation. 

II. A recognition of this need in current pedagogical literature 

and syllabi . 

HI- What has been done in the way of solving the problem, and the 
method of attack. 

1. The work and results of Phillips. 

2. The work and results of Griggs. 

3. The work and results of During. 

IV. An estimate of these investigations—their limitations in scope 

and method . 

V. The purpose of this study. 

VI. The plan of the investigation. 

1. Difficulty as shown in learning the combinations. 

2. Difficulty as shown by errors made in the combinations... 

3. Difficulty as shown by the combinations forgotten. 

4. Difficulty as shown by the time required to write the com¬ 

binations . 

VII. The limits of the field of investigation. 

1. The two fundamental processes... 

2. The forty-five addition combinations. 

3. The seventy-eight multiplication combinations. 

4. The necessity and advantages of teaching both direct and 

reverse forms . 

5. (a) Tests in addition. 

(b) Tests in multiplication. 

(c) Conclusion . 

VIII. Grouping . 

1. The addition combinations. 

2. The multiplication combinations. 

3. Advantages of the grouping selected. 

IX. Method of procedure in the first part of the investigation. . . . 

1. Preliminary tests. 

2. Directions to teachers for conducting drills and tests. 

3. Perception cards . 

4. The time and method of testing. 

( a ) Objections . 

(&) Advantages . 

5. Recording results . 

G. New group tests. 

7. Final tests . 


PAGE. 

7 

11 

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12 

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16 
16 
16 
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17 

17 

18 
IS 
19 

19 

20 
20 
21 
21 
22 
22 
22 
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23 

23 

24 

25 

26 
26 
27 

27 

28 
29 



































X. Psychological justification of the method of presenting the 

combinations . 

1. The number concept. 

2: Putting content into number and process symbols. 

3. Objective demonstration of number facts and processes. . . . 

4. Memory and habit. 

5. Association,^ judgment, &c. 

G. Repetition and retention. 

7. Extent of the appeal to the senses. 

8. Some experiments showing the effects of varying modes of 

repetition on the process of memorizing. 

XI. Difficulty . 

1. Its meaning.. 

2. Its cause . 

3. Its measure . 

XII. Results in the study of addition. 

1. Results in teaching the forty-five addition combinations 

(Table III) . 

2. Summary and co-efficients of difficulty determinations 

(Table IV) . 

„ 3. Graphic comparison of the learning co-efficients of difficulty 

for boys and girls (Graph I). 

4. Graphic comparison of total errors co-efficients of difficulty 

for boys and girls (Graph II).,. 

5. Discussion of results. 

(a) Interpretation of Table IV. 

(b) Method of studying the graphs. 

(c) Analysis and significance of the graphs. 

'XIII. Results in the study of multiplication. 

1. The results in teaching the seventy-eight multiplication 

combinations (Table V). 

2. Summary and co-efficients of difficulty determinations 

(Table VI) ...'. 

3. Graphic comparison of the learning co-efficients of difficulty 

for boys and girls (Graph III). 

4. Graphic comparison of the total errors co efficients of diffi¬ 

culty for boys and girls (Graph IV). 

5. Discussion of the results in multiplication. 

XIV. Relative difficulty as indicated by errors alone. 

1. Plan .,.. 

2. Nature of the tests. 

3. Directions for teachers. 

4. Results— 

(a) Addition (Tables VII and VIII). 

1. Discussion .. 

2. Errors in process (Table IX), with discussion.... 

3. Graph V, with discussion. 

(b) Multiplication (Tables X and XI).>.. 

1. Discussion .;. 

2. Errors in process (Table XII), with discussion... 

3. Graph VI, with discussion. 


29 

29 

30 

30 

31 
31 

33 

34 


34 

36 

36 

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41 

44 

45 

46 

46 

47 
47 

49 

50 

51 
54-a 


54-b 

55 

56 

56 

57 

58 

59, 61 
61 
62 
63, 64 
65-67 
67 
69. 70 
70-a 








































0 


PAGE. 

XV. 1. The effect of summer vacation upon memory for the com¬ 


binations . 71 

(a) Addition (Tables XIII and XIV. and Graph VII) ... 72. 73. 74 

1. Discussion . 75 

(b) Multiplication (Tables XV and XVI, and Graph 

VIII) .76. 77, 77-a 

1. Discussion . 77 

XVI. Vacation schools. 

1. Results . 78 

XVII. Relative difficulty as indicated by skill in manipulation. 79 

1. Plan and nature of the tests. 79 

2. Object of the test. 80 

3. Manner of conducting the tests. 80 

4. Method of evaluating results. 81 

5. Results— 

(a) Addition (Tables XVI and XVII, and Graph IX)... 82-S6 

1. Discussion . 87 

(b) Multiplication (Tables XVIII. XIX, XX and XXI. 

and Graph X). 8S-96 

1. Discussion . 97 

XVIII. Conclusions . 98 

XIX. Suggestions growing out of the investigation. 99 

XX. Bibliography and references. 100 

















































































































































































































































































































































































































































































































































































































I. THE NEED OF SUCH AN INVESTIGATION. 


Hundreds of thousands of children throughout the civilized 
world are each year confronted with the task of mastering the ele¬ 
mentary facts of number. These number facts or processes are 
the result of centuries of development. They are now an indis¬ 
pensable part of human knowledge. They, therefore, constitute 
an absolutely unquestioned part of every system of elementary 
education. Any additional knowledge that may tend to simplify 
or to facilitate the mastery of these facts must be of value to the 
great army of teachers upon whom devolves the task of teaching 
them, and of even more value to the greater army of children whose 
task it is to learn them. It is helpful for both teacher and pupil 
to know where the points of difficulty lie, because it stimulates both 
to more intense activity when these points are reached. This ac¬ 
tivity finds expression on the part of the teacher in the invention 
of devices for teaching, in the development of methods of instruc¬ 
tion. With the pupil this activity manifests itself under proper 
guidance in greater attention, in greater effort to get and hold the 
fact to be acquired or the problem to be mastered. Like one about 
to perform a feat of strength, he rouses the energy which he thinks 
may be sufficient to meet the demands of the occasion. We have, 
therefore, sought by this investigation to determine the relative 
difficulty of the elementary combinations, recognized generally 
as such in the United States, of the two most fundamental pro¬ 
cesses in arithmetic, namely, addition and multiplication. We 
do not presume that this study is in any way commensurate with 
the far-reaching application of the problem involved. We can only 
claim that what follows describes the most comprehensive investi¬ 
gation of this particular field of inquiry that has been made up to 
the present time. 


8 


It is not an uncommon thing in the school room and out, when 
the combinations are learned by “tables,” to find little or no dis¬ 
tinction in the emphasis placed upon the various facts of a given 
table. And the same is sometimes true even among the various 
tables themselves. Two plus three receives just about as much at¬ 
tention as does 2 + 7; and 2X1 receives about the same em¬ 
phasis as does 2X9, though in both cases the difference in diffi¬ 
culty is at once apparent. The most serious objection to this 
method of procedure is the woeful waste of time involved in re¬ 
peating again and again facts that need but few repetitions to 
make their recall automatic. Xor can it be said that the writers 
of text-books of primary arithmetic have kept the matter of relative 
difficulty sufficiently in mind. Their chief aim seems to have been 
a logical development, perhaps a psychological development, of all 
the elementary number facts in their regular order, and the giving 
of comprehensive miscellaneous review^ drills from time to time as 
the, progress of the work would seem to demand. To the credit of a 
few be it said that they have designated, somewhat indefinitely, 
to be sure, a few of what might be deemed difficult combinations, 
but they have, for the most part, laid the burden of determining 
the emphasis necessary to their mastery upon the teacher. This 
is doubtless only partly as it should be. The writer of such a text¬ 
book should know rather definitely beforehand just where inherent 
difficulty lies, so that he may present his material accordingly, and 
the teacher should have some more specific guide than is furnished 
in the ordinary text as to the place and amount of emphasis neces¬ 
sary for best results. She should know about what results she may 
reasonably expect in teaching a given group of number facts with a 
given amount of drill. It is the object of the present investigation 
to furnish, as far as may be, just this information. 

In order to determine just what emphasis was given the various 
combinations in addition, as shown by frequency of recurrence, 
the writer made a detailed study of three of the best number 
primers at present on the market. (See Table I.) These books 
were written with the idea of being put into the hands of the child 
at the end of the first two or three weeks of his school experience, 
and suggesting sufficient material to occupy his attention until 
about the end of the first half of the second year. 


9 


TABLE I. 


SHOWING THE NUMBER OF PRESENTATIONS GIVEN TO EACH OF THE ADDITION 
COMBINATIONS IN THREE OF THE BEST NUMBER PRIMERS. 


6 + 3 

7 + 3 

8 + 3 

9 + 3 

4 + 4 

5 + 4 

6 + 4 

7 + 4 

8 + 4 

9 + 4' 

5 + 5 

6 + 5 

7 + 5 

8 + 5 

9 + 5 
6 + 6 

7 + 6 

8 + 6 
9 + 6 

7 + 7 

8 + 7 

9 + 7 
8 + 8 
9 + 8 
9 + 9 


66 

55 

42 

43 
42 
72 

34 

35 
39 
33 

36 
33 
30 
25 
18 
11 
15 
10 
11 

3 

9 

9 

5 

11 


60 

38 

21 

30 

33 

45 

36 

20 

23 
18 
47 
29 

24 
27 
20 
11 
27 
17 

15 
8 

19 

19 

7 

16 
9 


nbination. 

No. 1. 

No. 2. 

3. 

0 + 0 I 

1 

76 1 

5 

1 + 0 

12 

24 

49 

2 + 0 

27 

47 

47 

3 + 0 

17 

33 

50 

4 + 0 

13 

29 

55 

5 + 0 

14 

50 

38 

6 + 0 

5 

31 

47 

7 + 0 

5 

30 

34 

8 + 0 

3 

38 

41 

9 + 0 

5 

26 

27 

1 + 1 

44 

S8 

42 

2 + 1 

85 

140 

114 

3 + 1 

80 

72 

104 

4 + 1 

63 

60 

95 

5 + 1 

44 

58 

92 

6 + 1 

38 

41 

93 

7 + 1 

30 

55 

79 

8 + 1 

36 

62 

78 

9 + 1 

20 

24 

61 

2 + 2 

71 

47 

65 

3 + 2 

130 

114 

134 

4 + 2 

105 

71 

13S 

5 + 2 

89 

75 

109 

6 + 2 

61 

75 

128 

7 + 2 

59 

53 

103 

8 + 2 

48 

41 

92 

9 + 2 

40 

26 

94 

3 + 3 

58 

46 

45 

4 + 3 

109 

72 

119 

5 + 3 

98 

75 

103 


Total Presentations.! 


2105 


2298 


93 

88 

76 
78 

35 

98 

99 
SO 
90 

77 
49 
83 
72 
68 
76 
39 
98 
85 
71 

36 
17 
66 
38 
69 
20 


3982 































10 


In determining the number of presentations of the various com¬ 
binations, continuous counting by 2’s, 3’s, 4’s and 5’s to a certain 
sum was omitted. Primer No. 1 thus gave 2,105 presentations; 
No. 2 gave 2,298 presentations, and No. 3 gave 3,982 presenta¬ 
tions. A comparison of the corresponding columns of Table I 
shows an interesting lack of uniformity in the relative number of 
presentations of the x facts involved. If the number of presenta¬ 
tions given by the three authors to the various combinations is to 
be regarded as a measure of the relative difficulty of these combina¬ 
tions, then No. 1 would have us believe that 3 —J— 2 is the most 
difficult fact to fix in the pupil’s mind, and that 0 + 0 is the least 
difficult. No. 2 would have us believe that 2 -f- 1 is the most diffi¬ 
cult and needs 140 repetitions during the time given to fix it in 
the child’s mind, while 8 —8 is the least difficult and needs but 
7 repetitions to accomplish the same result. No. 3 says that 
4 + 2 is the most difficult and will require 138 repetitions, while 
the child can learn 0 + 0 in 5 repetitions, and 9 + 9 in 20 repe¬ 
titions. Considering the total number of presentations given, we 
should expect No. 3 to give approximately twice as many repeti¬ 
tions of each combination as No. 1, yet they give 1 + 1 practically 
the same number of times, while No. 2 gives twice as many repeti¬ 
tions of this fact as either of the others. No. 2 would indicate that 
0 + 0, 3 + 1, 4 + 2, 5 + 2, 0 + 2, 4 + 3, and 5 + 3 are of 
practically the same difficulty, while the difference in difficulty 
in these combinations as shown by No. 1 and No. 3, varies con¬ 
siderably. The fewer repetitions of the double numbers; as, 
7 + 7, with No. 3 and No. 2 may be attributed not to a belief 
that they may be easier than others, but to the fact that there 
are no reverse forms, as with 7 + 5, which also is made to occur 
as 5 + 7. There is a double significance in these comparisons: 
(1) there is no agreement among authors as to the proper num¬ 
ber of presentations to be given to the various facts; and (2) the 
number of repetitions of a given combination by the same author 
is in no sense a guide to the teacher in determining the relative 
difficulty of that combination. 


11 


II. RECOGNITION IN CURRENT PEDAGOGICAL LITERA¬ 
TURE AND SYLLABI OF THE NEED OF SUCH 
AN INVESTIGATION. 

That there is in this matter of relative difficulty of number 
combinations in both addition and multiplication a problem worthy 
of investigation has been pointed out by such specialists in edu¬ 
cation as Prof. Henry Suzzalo (31), Dr. A. Duncan Yocum 
(44), and Dr. C. W. Stone (36) ; and indeed the best teachers 
of primary number work have for a great many years recognized 
its existence, and have acted upon their more or less satisfactory 
opinions and observations in their teaching practice. Certain 
investigations along this line have even been made, in this country 
by Phillips (23), Griggs (5) and Phelps (22), and in Germany by 
During (2), the methods and results of which we shall take the 
liberty of giving and discussing in subsequent pages. 1 Such writ¬ 
ers on special method as McMurry (17), Yocum (46), Quigley 
(26), Kigier (28), and Miss Gildemeister (3), have each made 
important suggestions arising out of their recognition of a dif¬ 
ference in difficulty of either addition or multiplication combina¬ 
tions, or both. So also has one of the most recent state mono¬ 
graphs (21) on the teaching of arithmetic this to say: “Much 
waste of effort will be avoided if these facts (of addition) are 
separated into groups according to their difficulty,” . . . and 

again, ‘‘the difficulties in addition are largely because a few simple 
combinations are troublesome such as 9 —|— Y, 8 + 5, 8 + 7, 
7 + 4, 5 + 2, 5 + 3, 8 + 5. The stress of drill should be 
placed upon these and other points of difficulty.” Doubtless much 
other evidence might be found to show a still wider recognition 
of the existence of an important problem here, hut that which is 
cited above seems quite sufficient for our purpose. 

1 P. Ranschburg (27), of Budapest, in his articles zur physiologischen und 
pathologischen Psychologie der elementaren Rechenarten, referred to by Ernest 
C. McDougle (15) in his excellent summary of studies made in arithmetic as 
treating the problem of “relative difficulty,” deals with the relative difficulty 
of the four fundamental processes, comparing the results obtained from normal 
and backward children. 



12 


HI. WHAT HAS BEEN DONE TOWARD THE SOLUTION OF 
THE PROBLEM, AND THE METHODS OF ATTACK. 

1. THE WORK AND RESULTS OF PHILLIPS. 

The earliest attempt reported in this country to gather any 
systematic information along the line of our inquiry was made in 
1897 by Mr. P. E. Phillips (23), then of Clark University. Mr. 
Phillips sent out a syllabus to 616 persons, 72% of whom were 
teachers, and the other 28% men and women whose occupations 
are not designated. Among a number of other questions this 
was asked: “What numbers give most trouble in adding or mul¬ 
tiplying?’’ There were 440 returns received, 157 of which gave 
7 and 9; 88, 7 and 8; 34, 6 and 7; 42, 7 only; 26, 3, 6 and 8. 
The others were miscellaneous. Seven is found in 327 cases, 9 
in 204. Five say that 9 is easy, always one less than 10. In ad¬ 
dition to this, the errors in multiplication found in the papers of 
283 ninth grade pupils involving 1,095 problems or multiplica¬ 
tions, the majority of which, contained three figures in the mul¬ 
tiplier and four in the multiplicand, were tabulated and are 
worthy of note. There were in all 691 mistakes in multiplication. 
Of these 186 were made in multiplying by 9; 195 by 8; 199 by 
7; 57 by 6; 9 by 5 ; 15 by 4; 3 by 2. (24). 


2. THE WORK AND RESULTS OF GRIGGS. 

Mr. A. O. Griggs (5), also of Clark University, in an article 
published in the Pedagogical Seminary, September, 1912, en¬ 
titled “The Pedagogy of Mathematics,” has done us the service 
of bringing together the work of Mr. Phillips and that of Pro¬ 
fessor Poring. Where there is a difference of results, he allows 
those of Professor Poring the preference for reasons that will 
appear in the following summary of the latter’s experiments. 


13 


3. THE WORK AND RESULTS OF DORING. 

Prof. Max Doring’s investigation (2) is one of two very 
satisfactory pieces of work that have been done up to the pres¬ 
ent time to throw more light on this particular problem. 1 His 
study involved only the multiplication combinations up to 
10 X 10. He started out by asking a number of adults (mostly 
teachers, both men and women) to name the three elementary 
multiplication combinations which in the opinion of each, based 
upon his own private and his pedagogical experience, were the 
most difficult. The results were so surprisingly uniform that he 
extended the inquiry to 323 boys in age from nine to fourteen 
years. To each of these the same question was put in writing, 
and after some deliberation each wrote down his opinion. To 
avoid the possibility of their hesitating to put down their real 
difficulties, they were not required to write the answers to the 
combinations which they might give. 

In compiling the results, the numbers with their reverses, as 
7X8 and 8X7, were tabulated under the same heading of 
8 X 7 as one fact. Sixteen combinations appear in 880 out of the 
1,053 opinions given. Sixteen and four-tenths per cent, of the 
opinions were eliminated as being unworthy of statistical ex¬ 
pression. Forty-nine per cent, of the opinions indicated 8X7 
to be the most troublesome combination, 34% stood for 9X7, 
30% for 9X8, 30% for 7X6, 21% for 9 }< 6, &c. That the 
selection of the combinations indicated was not a mere chance 
selection is demonstrated by statistical formulae for determin¬ 
ing the mathematical probability and the possible variation there¬ 
from. 

From the sixteen combinations, Professor Doring next pro¬ 
ceeded to arrange the various tables in the order of difficulty 
suggested thereby. Hoting that the factor 7 appears most fre- 

1 C. L. Phelps of Leland Stanford University (22) in 1913 worked over from 
a different point of view material collected by Otis and Davidson [“The Re¬ 
liability of Standard Scores in Adding Ability,” El. School Teacher, Oct., 1912] 
from 238 eighth grade pupils to determine the frequency of error in addition 
combinations. For his results partially stated, see Table VIII. 



14 


quently in the opinions tabulated, he concluded, as Dr. McMurry 
also states (18), that the seventh row is the most difficult. For 
the sake of comparison we give below the order of difficulty thus 
obtained side by side with that suggested by Dr. McMurry, Miss 
Gildemeister (4), and Professors Pocke, Poegers and Wolf (13). 
The most difficult tables are placed on the left, the least difficult 
on the right of the table. 


TABLE II. (a). 

Doring . 7 8 9 6 4 5 3 10 2 1 

McMurry . 7 9 6 3 8 4 5 2 10 .. 

Gildemeister (omitting 0, 11 and 12). 8 7 6 4 3 5 9 10 1 2 

Roclce, Roeger & Wolf. 7 9 6 3 8 4 5 10 2 1 

This is indeed an interesting comparison of opinion and ex¬ 
perience with the simplest possible method of statistical investi¬ 
gation. Professor Doring recognizes three general orders of dif¬ 
ficulty: the 1st, 2d and 10th tables he regards as “leicht”; the 
3d, 5th and 4th, as “mittel schwer”; and the 6th, 9th, 8th and 
7th, “schwer.” 

In order to check up his first results this investigator selected 
the ten most difficult combinations as determined by his first ex¬ 
periment and gave them as a test to 100 boys of the third school 
year. Among the 9,600 answers appeared 175 errors, or 1.8% 
distributed as follows: 


TABLE II. (b). 

New Place.. 1 *2 34567 89 10 

Combination, 9x7 9x6 8x8 8x7 8x6 7x6 9x8 7x7 8x4 9x9 
No. Errors.. 25 25 23 21 21 17 14 13 12 4 

Old Place... 25716438 10 9 

The number of errors is, of course, so small as to deprive this 
part of the experiment of any conclusive results, yet it is inter¬ 
esting here to note the comparatively small divergence from the 
first order given, and this is all the more interesting when it is 
noted that special search is made in the third year of the German 
schools for the difficult combinations, and special drill is given 
upon them. 






15 


IV. AN ESTIMATE OF THESE INVESTIGATIONS—THEIR 
LIMITATIONS IN SCOPE AND METHOD. 

If nothing else were accomplished by the contributions de¬ 
scribed above than to call the attention of the teaching profes¬ 
sion to the existence here of an important factor in the acquisi¬ 
tion of number facts, they would even then be eminently worth 
while. But they have done more. In the first place they have 
suggested methods of attacking other problems of no less im¬ 
portance in school-room practice. In the second place they have 
shown that, while empiricism may be a reliable guide to such 
practice, it should be verified by careful experiment and receive 
the sanction of scientific investigation. Only thus can certainty 
and dignity be given to method. And, lastly, they have con¬ 
tributed information of real value in acquiring mastery over a 
very essential part of the school curriculum. 

If these things are true, how shall we justify further investi¬ 
gation along the lines indicated? It will, in the first place, be 
noted that the relative difficulty of the addition combinations is 
scarcely touched upon, except for eighth grade pupils (see foot¬ 
note, page 13), while practically only sixteen of the multipli¬ 
cation combinations are statistically considered. The limitations 
first noted are justified in the light of the larger subject of “Num¬ 
ber and its Application,” discussed by Mr. Phillips (23), so 
likewise with that of “The Pedagogy of Mathematics,” discussed 
by Mr. Griggs(5). But the limitations of Professor Boring’s 
study were set by himself rather than by his subject to the 
“difficult cases,” though his subject, “Zur Psychologie des Kleinen 
Einmaleins,” would suggest a somewhat more comprehensive view 
of the psychological side of the problem than he gives. 

Aside from the limited scope of the studies made, it is doubt¬ 
ful if the questionnaire method with adults or the “method of 
errors” in problems involving many figures in the multiplicand 
and multiplier, where some of the errors may be errors in “carry¬ 
ing,” are greatly to be relied upon. So likewise with a method 
which depends largely upon the “opinions” of those tested. The 
data obtained by the use of such methods should at least be checked 
up by results obtained by methods not open to the objections that 
may be offered to any of the above mentioned modes of procedure. 


/ 


16 


V. THE PURPOSE OF THIS STUDY. 

It is the purpose of this study to determine, as nearly as may 
be, the relative difficulty of certain number facts, and to express 
the data obtained in such a way that primary teachers may be 
furnished a more definite guide than now exists as to the emphasis 
that should be placed upon these facts in their attempt to im¬ 
part them to the child. It will also be a part of our task (to 
suggest some of the elements of method which may be profitably 
used to reach the ends of instruction in this field in a more ex¬ 
peditious way than has formerly been done. 


VI. THE PLAN OF THE INVESTIGATION. 

The teaching of both the addition and the multiplication com¬ 
binations requires, in the nature of the case, the breaking up of 
the work into certain groups, which must consist of fewer facts 
than are given in a single “table.” It is, therefore, important 
that the teacher know the points of difficulty in each of the several 
groups as they are considered. In other words, it is important 
that a survey of the whole field of elementary number facts be 
made rather than only the mountainous districts. Small climbs 
to the untrained beginner may tax his energies quite as much as 
the more inaccessible peaks do to him who by training and ex¬ 
perience has learned the art of the mountain climber, and has 
developed in the process the physical powers necessary to the 
bigger task. 

1. DIFFICULTY AS SHOWN IN LEARNING THE COMBINATIONS. 

Since children have to master the facts as presented, it has 
been our plan to study the difficulties which the children them¬ 
selves have experienced in the process. This, though probably 
the most difficult plan of attack—surely it has been the most 
tedious—has seemed to us to be the most logical. It is the child’s 
point of view that must in the end be most helpful for the teacher 
to get. 


17 


2. DIFFICULTY AS SHOWN BY ERRORS MADE IN THE COMBINATIONS. 

We have also attacked the problem from another side. We have 
determined the relative number of errors made by large numbers 
of children in writing the combinations after these children have 
been over the ground and were presumably familiar with them. 
This plan differs from Professor Doring’s second method of deter¬ 
mining his results only in the fact that we have in our experi¬ 
ments provided equal opportunity to make errors in all of the 
combinations, while he gave tests on only the ten combinations 
which he found most difficult in his first experiment. 


3. DIFFICULTY AS SHOWN BY THE COMBINATIONS FORGOTTEN. 

It also seemed wise to consider the relation between the various 
number facts from the standpoint of their permanence when once 
mastered. It is not infrequently the case that things easily 
learned are quite as easily forgotten. With this point in mind 
we have given tests in all the number facts to over 500 third grade 
children during the last week of their school work in June, and 
have given the same tests to the same children in the first week 
of their school work in September following. We have thus 
determined the facts which are most likely to drop out of mem¬ 
ory during the long summer vacation. This should be of interest 
to a fourth grade teacher just taking a class promoted from the 
third grade. 

4. DIFFICULTY AS SHOWN BY THE TIME REQUIRED TO WRITE THE 
COMBINATIONS. 

. And finally, since the more difficult a mental process is, the 
more thought it requires, the less skilful the individual is in its 
use, the greater is the time required in its accomplishment. It 
has, therefore, been also our plan*to determine the relative dif¬ 
ficulty of the various groups of facts as wholes by comparing the 
length of time required to write correct results to the various 
combinations of which these groups were constituted. These tests, 
like that described immediately above, were given only to pupils 
who had been taught all the facts involved. 


18 


VII. THE LIMITS OF THE FIELD OF INVESTIGATION. 

1. THE TWO FUNDAMENTAL PROCESSES. 

Of the “four fundamental processes” in arithmetic, two may be 
regarded, in the light of modern methods, as fundamental, and 
the other two derivative. Subtraction may be regarded as addi¬ 
tion from another point of view, as far as the process goes, if the 
Austrian or making change method is used, and division is like¬ 
wise multiplication from another point of view. In the final 
analysis there is really only one fundamental process, viz., addi¬ 
tion, all the others being derived from this. While multiplica¬ 
tion may be regarded as “shorthand” addition, and so taught, yet 
it is not the addition of a row containing seven 4’s that a child 
should see when he says seven 4’s are 28. The latter is purely 
an abstract process, entirely different psychologically from add¬ 
ing 4 and 4 and 4, &c., seven times. It is quite in accord with 
the best established pedagogical principles that the known process 
of addition be carried over and made the basis of its modifica¬ 
tion in multiplication, but its raison d’etre having been once 
pointed out the connection should cease as far as the process goes. 
During the process of multiplication introspection shows no con¬ 
scious association with the process of addition. This is not true in 
the case of subtraction, except when it has been taught as a separate 
process, nor is it the case with division, there being in the first 
case a conscious background of addition, and in the second a 
conscious background of multiplication. Four from twelve leaves 
eight, because eight and four are twelve; four into twelve goes 
three times, because three times four are twelve. For these 
re'asons we have selected what we regard as the two most funda¬ 
mental processes for the field of our investigation, and we believe 
that what may be found true in this field may be regarded in a 
very large measure as true in the related fields of subtraction and 
division within the same range. 


19 


2. THE FORTY-FIVE ADDITION COMBINATIONS. 

If all the possible combinations in addition from 0 to 9 are 
counted, it will be found that the total is 100. Of these, 19 are 
combinations involving 0. There are then 81 combinations of the 
9 significant numerals. Of these, 9 admit of no reverse forms; 
as, 1 + 1, 2 + 2, 3 + 3, &c. Excluding these 9 there are 72 
combinations, counting both direct and reverse forms; as, 
4 + 3 = 7, and 3 + 4 = 7. If we regard both of these forms 
as one number fact, there are in these 72 combinations 36 num¬ 
ber facts, which, added to the 9 admitting of no reverse form, gives 
us 45 fundamental addition combinations. In this investigation 
we have dealt with the 81 combinations in the teaching process 
for reasons which we shall presently explain, and we have done 
the same in the “time tests” (See VI 4), but in the final test to 
determine the number of errors made after the teaching process 
was finished only the 45 fundamental facts were given. Since 
the combination of a single number with 0 never occurs in prac¬ 
tice—we are never required to add 0 objects! to 4 objects or 4 
objects to 0 objects—the necessity appearing only in the handling 
of numbers above 9, as when we say 8 and 10, or 30 and 15, it 
was though unnecessary to include the 0’s in the experiments. 
All the results in addition are, therefore, subsumed under 45 
fundamental heads. 

3. THE SEVENTY-EIGHT MULTIPLICATION COMBINATIONS. 

Ill this country it is customary to teach the multiplication 
tables up to 12 X 12. It is also generally the custom not 
to teach the combinations with 0 as a part of the tables, they 
being left till they are required in column multiplication. The 
combinations with the ones, however, are taught, at least in the 
form in which the one appears in the second factor, or is used as 
the multiplicand. We hear, for instance, two l ? s, three 1 s, four 
l’s, &c., at the beginning of each table. Hot counting the com¬ 
binations using 0 as a factor, there are 144 combinations from 
1 X 1 to 12 X 12, inclusive. Of these 12 have no reverse form; 


as, 1 X 1, 2 X 2, 3 X 3, &c. Excluding these 12 there are 132 
combinations counting both the direct and the reverse forms; as, 
7 x 4 = 28 and 4 X 7 = 28. If we regard both of these as 
one number fact, there are in these 132 combinations 66 funda¬ 
mental facts, which, added to the 12 admitting of no reversions, 
gives us in all 78 fundamental facts in all the multiplication 
tables. While the multiplication combinations have been taught 
in both direct and reverse forms for reasons which we shall sub¬ 
sequently adduce, and the “time tests” were made to include 
both of these forms, as in the case of addition, only 78 facts were 
required in the final tests, and the results of all were subsumed 
under the 78 fundamental heads. 


4. THE NECESSITY AND ADVANTAGES OF TEACHING BOTH DIRECT 
AND REVERSE FORMS. 

One of the earliest problems which confronted us in conducting 
this investigation was whether we should take up the “tables” 
in addition and multiplication in the time honored way, or 
whether we should reduce ' the number of combinations to the 
forty-five fundamental addition facts and the seventy-eight fun¬ 
damental multiplication facts. Upon the decision on this point 
depended a most important factor in the method of presenting 
the work. If it could be shown that a pupil who knew perfectly 
4 + 7= 11, would know equally well 7 -(-4 — 11, or if it could 
be shown that a pupil who knew surely 4 X 9 = 36, would know 
with equal certainty 9 X 4 = 36, this would entirely eliminate 
the teaching of all reverse forms. 

a. Tests in Addition. —In order to determine what was pos¬ 
sibly true in regard to this matter in addition, a test was made 
with 29 children, 14 boys and 15 girls, in the lower half of the 
first grade. The experiment began on October 28th and was con¬ 
cluded November 1st, 1912. The children had been in school 
since the middle of September, and had already learned a few 
of the simpler addition facts. The following facts in the form 
indicated were selected for the drill: 6 + 8, 7 + 6, 9 + 8, 

8 + 7, 9 + 6. These facts w T ere drilled upon with equal oral 


and written emphasis from Monday till Thursday, inclusive. On 
Friday morning at ten o’clock, after sixty minutes in school, a 
written test was given on the forms as drilled. At eleven-twenty 
o’clock of the same morning, forty-five minutes after a twenty- 
minute recess and singing period, the same test was given in the 
reverse form. The results are given below: 

No. Total % 

Pupils. 6-|-8 7+6 9 + 8 8+7 9 + 6 Correct. Correct. 


Direct Form.. 29 29 29 27 29 27 141 97.2 

Reverse Form. 29 21 25 19 20 20* 105 72.3 


Out of a possible 145 correct results it will be noted that ten 
times as many errors were made in the reverse form as in the 
direct form test. 

b. Tests in Multiplication .—At the same time that the experi¬ 
ment in addition was being conducted a similar experiment in 
'multiplication was being carried on with 40 pupils, 20 boys and 
20 girls, in the lower half of the second grade. In this experi¬ 
ment there was only one-fourth as much written drill as oral drill. 
The final test in the direct form was given at ten a. m. on Novem¬ 
ber 1st, and that in the reverse form at eleven o’clock on the same 
morning, fifteen minutes after a ten-minute recess. The results 
are given in the following table: 

No. Total % 

Pupils. 3x4 7X3 6x4 5x3 4x5 Correct. Correct. 


Direct Form. 40 40 39 38 38 38 193 96.5 

Reverse Form. 40 40 33 35 37 35 180 90 


The selection of 3 X 4 as one of the combinations was doubtless 
not well made on account of its manifest ease, yet even in this 
experiment nearly three times as many errors were made in the 
reverse form test as in the direct form test. 

c. Conclusion ,—From these experiments we concluded that, 
while most of the direct form teaching could doubtless be carried 
over and applied to the reverse form, not a sufficient amount was 
transferred to justify the teaching of one form and omitting to 
teach its reverse, and since the time of Grube (1842), it may 
be said that the teaching of both forms has been in accord with 
the best pedagogical practice. We, therefore, have double justi¬ 
fication for the plan pursued in this investigation. 






VIII. GROUPING. 


1. THE ADDITION COMBINATIONS. 

The necessity of teaching both the direct and reverse forms 
naturally brought up the question as to the order in which they 
should be taught. This in turn involved still another problem, 
that of grouping the combinations to be taught. Should we take 
one table at a time in what might be regarded as the logical order, 
or should we take a psychological order, such as is pointed out 
by Dr. Yocum (46) for the addition facts, or one such as is 
indicated by Superintendent Rigler (28) in the Portland, Oregon, 
Course of Study, or what may be regarded as the promiscuous 
order suggested by the Los Angeles Course of Study (14). As 
none of these is even approximately the same, it was concluded 
to accept the psychological order suggested by Dr. Yocum’s ex¬ 
periment and break his groups into smaller ones or combine them 
into larger ones containing each five facts, the idea being that 
one new fact each school day for nine weeks might cover the 
ground in addition. As a result the following groups with their 
reverse forms were selected: 


1. 

1 + 1, 

2 + 1, 

3 + 1, 

4 + 1, 

5 + 1 

2. 

6 + 1, 

7 + 1, 

8 + 1, 

9 + 1, 

2 + 2 

3.’ 

3 + 2, 

4 + 2, 

5 + 2, 

6 + 2, 

7 + 2 

4. 

8 + 2, 

9 + 2, 

3 + 3, 

4 + 3, 

5 + 3 

5. 

6 + 3, 

7 + 3, 

8 + 3, 

9 + 3, 

4 + 4 

6. 

5 + 4, 

6 + 4, 

7 + 4, 

8 + 4, 

9 + 4 

7. 

5 + 5, 

6 + 5, 

7 + 5, 

8 + 5, 

9 + 5 

8. 

6 + 6, 

7 + 6, 

8 + 6, 

9 + 6, 

7 + 7 

9. 

8 + 7, 

9 + 7, 

8 + 8, 

9 + 8, 

9 + 9 


2. THE MULTIPLICATION COMBINATIONS. 

In multiplication the first nine facts were made to constitute 
the first group for those who had already had some work in the 
process, i. e., from 1 X 1 to 9 X 1. For those who were begin¬ 
ning the work for the first time these nine facts were divided 
into two groups, the first of five combinations, the second of four. 
The remaining combinations were divided into groups of five 
facts each, in the same order as shown above for the addition 


23 


facts, up to the last group, which consisted of only four facts, 
12 X 10, 11 X 11, 12 X 11, 12 X 12. There are thus sixteen 
groups of multiplication facts, it being the idea that this ground 
might be covered in sixteen weeks of five days each, thus giving 
an average of one new fact each day. Experience with this 
manner of grouping, and the definiteness of the work required 
made it possible to accomplish the tasks in the time allotted ex¬ 
cept with the class entering school in the fall term. The cor¬ 
responding class in the spring term, however, had no difficulty in 
keeping up with the schedule. 


3. ADVANTAGES OF THE GROUPING SELECTED. 

The advantages of this manner of grouping are two in num¬ 
ber: (1) the general idea of the sequence has already been shown 
to be psychological (46), at least as far as the addition facts are 
concerned; and (2) this order following the order of the numeri¬ 
cal scale does not lay the investigator open to the charge of a 
predetermined attitude of mind toward the problem of relative 
difficulty by a special grouping devised to demonstrate some pos¬ 
sible theory of his own. 


IX. METHOD OF PROCEDURE IN THE FIRST PART OF 
THE INVESTIGATION. 

1. PRELIMINARY TESTS. 

The first direct step in the investigation was made at the be¬ 
ginning of the second term, February 3d, 1913. It consisted in 
giving preliminary tests in addition to the first grade and the 
first half of the second, and in multiplication to the second half 
of the second grade and the third grade. The forty-five addi¬ 
tion combinations in promiscuous order were arranged in columns 
on a sheet of paper in the following form: 2 + 5 = , 

3 -f- 4 = , &c., with the answers omitted as indicated. Simi¬ 

larly, the seventy-eight multiplication combinations were arranged 
in promiscuous order in three general divisions, representing 
about the first, second and third third’s, respectively, of the whole 


24 


number of facts, tlie object being to arrange the test to suit in 
some degree the various stages of advancement of the pupils 
tested. The sheets on which the combinations had been mimeo¬ 
graphed were distributed face down, one on the desk of each 
pupil in the various rooms. The teachers then explained that 
there were a number of little examples in addition or multipli¬ 
cation, as the’case might be, on the sheet, and they wished to 
see how quickly each pupil could write the answers after the 
equal sign, illustrating on the blackboard just what was expected. 
The pupils were then told to turn their papers and write. They 
were permitted to write as many answers as they could. This 
test furnished the preliminary test record for each pupil for 
each combination studied until the completion of the drill. The 
same method was pursued at the beginning of the experiment in 
September, 1913, except in the first grade, where the pupils could 
write only with great difficulty, if at all. The results here were 
obtained by oral examination by the teacher. 

This done, each teacher w T as provided with a list of the groups 
to be taught as indicated in Section VIII, and a copy of direc¬ 
tions as follows: 

2. DIRECTIONS TO TEACHERS FOE CONDUCTING DRILLS AND TESTS. 

The same amount of time and the same emphasis should be used each day 
in presenting the groups of combinations to be taught. The objective demon¬ 
stration should be used only twice, and on successive days. The presentation 
should consist of ten (10) oral repetitions, three (3) of which should be in 
concert with the whole body of pupils studying the group, and the other seven 
(7) by sections and individuals. There should be five (5) written repetitions 
of each number fact each day. In all of this work of repetition only the correct 
form is to be allowed. 

Subtraction should be taught as the reverse process of addition, and division 
as the reverse process of multiplication. The direct and the reverse processes 
should go hand in hand. 

The multiplication facts may be taught in the form of tables, but they should 
appear in both the forms as shown below: 

2X1= 1X2= 

2x2= 2X2= 

2x3= and 3x2 = 

2x4= 4X2= 

2x5= 5X2= 

but all tests should be made promiscuously, both when oral and when written. 

Teach addition combinations whose sums do not exceed ten first by the use of 
objects, but only twice thus. 

Teach multiplication as short-hand addition, illustrating by the repetition 
of the same number in column arrangement to be added. 


25 


Drill after each written test until at least 97% of class efficiency is obtained, 
then proceed to the next group. 

Review all combinations finished twice each day, once oral and once written, 
preferably in connection with the problem work. 

Two weeks after giving the final test on each group a review test on that 
group is to be given along with the daily test on the same sheet of paper, but 
on the reverse side, such side to be marked “Review.” 

Read over these instructions carefully and follow them implicitly, as the 
results will be vitiated by any lack of uniformity in the presentation of the 
material. 


3. PERCEPTION CARDS. 


Iii order to facilitate the drills and tests and at the same time 
provide an easy means of visual instruction, special perception 
cards for both addition and multiplication were prepared bv the 
writer in the form indicated in the following figures: 


Fig. 1. 

4 + 7 = 11 
7 + 4 — 11 

Obverse. 


Fig. 2. 


4 

+ 7 


+ 4 


Reverse. 


Fig. 3. 


4 X 4 = 16 


Obverse. 


Fig. 4. 


4 

X 4 


Reverse. 





























26 



The side represented by Fig. 1 was the side used for drill. 
The side represented by Fig. 2 was that used for testing in addi¬ 
tion. A similar arrangement was used for multiplication. Fig¬ 
ures 3 and 4 above indicate the disposition of the numbers having 
no reverse form. The reason for the different arrangement of 
the numerals on the obverse and the reverse sides arises out of 
practical considerations. Figures 2 and 4 show the arrangement 
that is usually presented when problems are to be solved, while 
the arrangement shown in Figures 1 and 3 recommends itself for 
convenience in printing and in writing results. 

The teachers being already expert in the use of perception 
cards in the teaching of reading became quite as expert in a very 
short time in their use for teaching the combinations of number. 
These cards proved to be important savers of the teachers’ time 
and effort as well as effective means of visual instruction. 


4. THE TIME AND METHOD OF TESTING. 

At the beginning of the number period each morning a test 
involving the facts of the group drilled upon the day before was 
given; hence, there was always at least one day’s interval be¬ 
tween the drill and the test upon the group under study. Fre¬ 
quently, of course, a much longer time elapsed, as when a test 
came after a single holiday, when there was a two-day interval, 
or w T hen it came on Monday, thus giving a three-day interval, &c. 
For the tests the teacher either wrote the combinations on the 
blackboard, or, which was the usual method, arranged the per¬ 
ception cards containing the proper combinations in the chalk 
tray of the blackboard. The pupils copied these in the order of 
their arrangement and wrote under each the answer. These papers 
were then collected and the results recorded by the teacher. 

a. Objections .—It is recognized that two objections may be of¬ 
fered to the written test as a means of gathering data of this kind, 
especially with first grade children: (1) the difficulty that is 
experienced by them on the mere mechanical side of writing 
the symbols is likely to detract from accuracy in the process in¬ 
volved ; and (2) an increased opportunity to obtain answers by 


27 


‘‘counting up” in the case of addition, or “running down” the 
tables in multiplication. On the other hand, it was a matter of 
observation that children learned very quickly to express them¬ 
selves by means of the written symbol. For the most part they 
had come into the first grade from a kindergarten and reception 
grade where they had been taught to count, to recognize numbers, 
and to make the symbols on the blackboard. They had also had 
some practice in the use of the pencil for this purpose. As to 
“counting up” and “running down” the rows to reach the desired 
point, everything in the method of presentation and the teacher’s 
attitude was made to discourage those devices. 

b. Advantages .—The written method has the positive advan¬ 
tage of being free from the excitement or the stimulus of per¬ 
sonal contact with the questioner. It may be said to represent 
the calm judgment of the individual tested, just as the written 
productions of mature persons represent a better quality of thought 
and expression than their unprepared oral efforts. But the great¬ 
est advantage of this method was its adaptability to the purposes 
which it served. It would have been impossible without serious 
interference with the other work of the school for the teacher 
to have given an oral examination to each of her pupils in each 
of the facts of each group apart from the other members of the 
class each day of the week. 

5. RECORDIN'G- RESULTS. 

For recording results a special blank was prepared. This 
blank contained the entire record of a class throughout its work 
on a given group, from the preliminary test to the “review” test, 
two weeks after the last intensive drill. Since positive results 
from the standpoint of teaching were desired, a standard of at 
least 97% class efficiency being the aim in each group, the totals 
and percentages were based upon the correct answers in the 
various tests. As the results were all to be subsumed under the 
forty-five addition and the seventy-eight multiplication combina¬ 
tions, an error in either the direct or the reverse form or both 
was counted simply as one error, and a “X” was marked in the 
proper place to indicate the same. Correct results, were indicated 


28 


by a check mark (V)> The percentages each day were based 
upon the number of pupils present. By this method the teacher 
had before her each day the standing of the individuals, the 
standing of the combinations themselves, and, in the summary, 
the class accomplishment. It is interesting to note here that after 
the tests were all completed the teachers had found the scheme so 
helpful that many of them requested that they might be allowed 
to continue the use of this record. 


6. NEW GROUP TESTS. 

When all of the groups of the combinations had been taught 
and the “review” test on the last one had been given, the addi¬ 
tion combinations were divided into three groups, and a test was 
given on each of these new groups. After each of such tests was 
given, the five combinations in which errors were most frequent 
were again taken up for intense drill and dealt with in the 
same manner as originally, under the heading “second time 
over” in the final summary of results If frequent errors ap¬ 
peared in more than five combinations, then the other com¬ 
binations in which errors appeared were also given intensive 
drill, and so on until the difficulties had been covered a second 
time, unless the close of the term prevented, as was the case in 
a few instances. 

The multiplication combinations were similarly divided, but 
into five groups. These groups both in addition and multiplica¬ 
tion contained for the most part both the direct and the reverse 
forms of the number facts. Here again the combinations showing 
most errors were again taken up and drilled intensively. 

Unfortunately the results of those tests were not individualized, 
that is, were not placed in the individual records of the pupils 
taking them. Consequently they could not be included in the 
general summary of results because of the fact that a great num¬ 
ber of the pupils had to be eliminated from the results in various 
groups owing to absence. The records of the “second time over” 
drills would seem to indicate that most of the errors made in 
these tests must.have been made by the pupils thus eliminated. 


7. FINAL TESTS. 


At the end of the term final tests in the forty-five addition and 
the seventy-eight multiplication facts were given with the same 
kind of test sheets as were used in the preliminary tests at the 
beginning of the term (See IX, 1). The results of these tests 
were tabulated for each pupil and appear for the pupils not elim¬ 
inated in the final summary sheets (See Table III) under the 
head of “Final Tests.” 


X. PSYCHOLOGICAL JUSTIFICATION OF THE METHOD 
OF PRESENTING THE COMBINATIONS. 

1. THE NUMBER CONCEPT. 

To name things is an inherent tendency of the human mind. 
To represent these names by means of symbols is the next step 
in the natural course of its development—a step that may have 
required many centuries to take. No conscious reflection, no 
analysis, no synthesis is necessary to the process of naming. A 
child calls water “maw,” as the writer once knew a child to do. 
Aside from the conventions of language, his name for this sub¬ 
stance was quite as good, and did serve the purposes of his limited 
environment quite as well, as the word in common use. The 
symbol, by wdiich we mean the written symbol, is the product of 
reflection, of analysis, of synthesis. The symbol must of neces¬ 
sity have some relation, either real or fancied, either to the object 
or to its name. Numbering differs from naming in that it arises 
not from a perceptive process, which is of one thing or more than 
one as far as numbering goes, but, like the process of developing 
symbols, it arises out of reflection about things. Presumably 
people who have small powers of reflection, such as the savage 
races, will have very few number concepts. And that this is the 
case has already been pointed out by Branford (1). According 
to McLellan and Dewey, “Number is not (psychologically) got 
from things, it is put into them” (16). Nor yet is it gotten with- 


30 


out things. ‘So that to arrive at number concepts without the aid 
of things would be as impossible as to arrive at the concept “Greek 
Education” without the aid of symbols. 


2. PUTTING CONTENT INTO NUMBER AND PROCESS SYMBOLS. 

Content is put into symbols of number by showing their rela¬ 
tion to the concrete things for which the symbols stand, whether 
it be by measuring (1), or by counting (24), or by ratio (34), 
it matters little, adequate number concepts having been obtained 
in all of these ways. Content is put into symbols of process by 
showing with things the relations indicated by such symbols. To 
make clear the content of number and process symbols by the use 
of objects was the object of the two objective presentations of 
each combination within the limits set forth. 


3. OBJECTIVE DEMONSTRATION OF NUMBER FACTS AND PROCESSES. 

The number of objective presentations was limited to two. 
This was done to avoid the possibility of forming a habit of de¬ 
pending upon objects and using the methods of counting to obtain 
results. It was thought that this number of objective presenta¬ 
tions would be sufficient to demonstrate the meaning and purpose 
of the process involved. On this point Dr. Smith has the follow¬ 
ing to say: “It is important to use objects freely wherever they 
assist in understanding number relations, but it is equally im¬ 
portant to abandon them as soon as they have served their pur¬ 
pose. * * * To continue to use objects after they have ceased 
to be necessary is like always encouraging a child to ride in a 
baby carriage” (32). 

The number of objects used in teaching addition facts was re¬ 
stricted to ten for the reason that more than that number tends 
to confuse and encumber the process rather than to clarify it, 
while the pupil not being able to visualize the larger groups is 
forced into the very method which we strive to avoid—the count¬ 
ing by ones (42). 


31 


4. MEMORY AND HABIT. 

The facts of a given group when once developed make some im¬ 
pression on the mind of the child, but that impression is in¬ 
sufficient for the purpose of instruction, which is, that each fact 
should become a permanent and instantly available part of his 
mental equipment. The reaction to the abstract stimulus 3 + 4, 
3 

or + 4 should be “7” without the slightest hesitation, just as the 
reaction to m-a-n when seen or spoken is at once “man.” To 
obtain this result the laws of memory and habit must be invoked, 
for the correct reactions must not only be memorized but they 
must be made habitual. The most important mechanical factor 
in the production of both these results is repetition (11). There 
must be a great deal of repetition of the combinations in the ab¬ 
stract, repetition freed from objective paraphernalia in the usual 
sense, but repetition which may make use of various number de¬ 
vices which lend interest to the exercise. We may be pardoned 
for again quoting Dr. Smith. “It is a serious error,” he says, 
“to neglect abstract drill work in arithmetic. So far as scientific 
investigations have shown, pupils who have been trained chiefly 
in concrete problems to the exclusion of the abstract work are 
not so well prepared as those in whose training these two phases 
of arithmetic are fairly balanced. ... At the same time it 
(abstract computation) is the most practical part of arithmetic, 
since most of the numerical problems we meet in life are sim¬ 
plicity itself as far as the reasoning goes; they offer difficulties 
only in the mechanical calculations involved, and constantly sug¬ 
gest to us our slowness and inaccuracy in the abstract work of 
adding, multiplying and the like. In the first grade this work 
is largely but not wholly oral” (33). 

5. ASSOCIATION, JUDGMENT, ETC. 

Memory for facts can, of course, be aided by richness of asso¬ 
ciation, by judgment, by emotional appeal, and by voluntary at¬ 
tention. While all of these may and should play some part in 


32 


memorizing number combinations, the opportunities for their use 
for this purpose as compared with similar opportunities presented 
by poetry, or facts of history or geography, &c., are few. On this 
point, it is interesting to note Dr. McMurry’s observations. He 
says that “advantage should be taken of the similarity between the 
2’s, 4’s, and 8’s, also between the 3’s, 6’s, and 9’s. Note also the 
fact that in counting by 8’s the right-hand figure expresses two 
less each time, and in counting by 9’s one less. The children are 
interested and curious about these things, and they aid the mem¬ 
ory. The number 24 equals 4X6, 3X8? and 2 X 12. Such 
cases should be examined, and reasons given for the variety of 
factors in the same product. 

“In learning the multiplication tables later, the memory can 
be aided by a variety of these reviews, comparisons and rational 
analyses. 

“First of all, additions based upon objective work and meas¬ 
urement stand in the background of thought, as giving meaning to 
multiplications and divisions. Secondly, the repetitions and regu¬ 
larities running through some of the tables should be studied as 
of curious interest, as in the case of the 10’s and 5’s, 4’s and 8’s. 
Third, the identity of certain products in different tables, as 
4 X 5 = 20, and 5 X 4 = 20, and 2X10 = 20. A comparison 
and analysis of these identities is excellent thought work and a 
positive aid to the memory” (18). All these things are true, but 
they alone cannot be relied upon to give the child that facility in 
the use of the number combinations which will make him adept in 
their promiscuous use, in which form only they will be required 
of him in their practical applications. 1 

It was noted in teaching the addition combinations that when 
a double number, as 4 + 4, was mastered it formed a valuable 
connection with 4 + 3, and 4 + 5, the former being one less and 
the latter one more than the sum of the double number. This 
principle was also sometimes extended to sums two more or two 
less than the sum of the double number. Then, too, after drill 

1 The fact that 56=7x8, gives the natural number sequence of 5, 6, 7, 8 may 
be helpful, but the possibility of error is still large. The scheme for remem¬ 
bering the table of 9’s is interesting and should be pointed out to the learner, 
but not relied upon to take the place of necessary drill. 



33 


there was established a sense of fitness of answer to combination. 
One day the writer was in a 1—B grade (lower half) when the 
teacher at his suggestion called one of her boys to the desk and 
said, “Herbert, what are 8 and 5 ?” There was a moment’s hesita¬ 
tion, then came the answer “13.” “What were you doing while we 
were waiting for the answer ?” asked the writer. The reply was, 
“I was thinking over the numbers.” This doubtless meant that 
he was going over the number series until he came to the one that 
“fit” 8 + 5. Professor Thorndyke calls all such combinations 
“paired associates” (37). Given the stimulus 4X7, the neural 
discharge is into “28,” just as the stimulus of the German word 
“Gedachtnis,” for instance, discharges into “memory” for the 
English student of that language. For a great many of the com¬ 
binations the possible associations are just as remote as the relation 
between Gedachtnis and memory to one who knows no German, 
while in others the relations are so intricate that to remember the 
relation is more difficult than to commit to memory the combina¬ 
tion. 


6. REPETITION AND RETENTION. 

Dr. Yocum says, “The dominant factor of any method which 
has for its end maximum certainty and readiness of recall of the 
subject-matter to be taught is repetition” (47). Mr. Speer de¬ 
clares, “The way to succeed (in memorizing the tables, and he be¬ 
lieves in taking only a few of the facts of each at a time) is to 
develop vivid mental pictures, and to fix these pictures by bringing 
them again and again before the mind” (35). Dr. W. T. Harris 
went so far as to say that, “Lists of names, * * * also num¬ 

bers, as, e. g ., the multiplications, the melting points of minerals, 
&c., must be learned without aid. All indirect means only serve 
to harm here, and are required as self-discovered devices only in 
case that interest or attention has been weakened” (6). Dr. Ros- 
enkranz.himself declares (and the above opinion of Dr. Harris is 
a comment on his belief), “The means to be used (and these are 
based on the nature of memory itself) are, on the one hand, the 
pronouncing or writing the names or numbers, and, on the other, 
repetition; by the former we gain distinctness, by the latter sure¬ 
ness of memory” (7). 


34 


7. EXTENT OF THE APPEAL TO THE SENSES. 

It will be noted that, while repetition is the dominant feature 
of the drill given, this repetition involved as many of the senses 
as possible. First, it was made to appeal to the eye in the case of 
objective development, then further tO' the sense of vision as sym¬ 
bols by use of number perception cards specially prepared; second, 
to the motor and auditory senses in the oral drill, then further to 
the visual and motor sense in the written repetitions. Finally, 
teachers were instructed in general talks to make use of the in¬ 
stincts of play and of rivalry to accomplish results. 


8. SOME EXPERIMENTS SHOWING THE RESULTS OF VARYING MODES 
OF REPETITION IN THE PROCESS OF MEMORIZING. 

In this connection it is interesting to note some recent experi¬ 
ments in committing to memory that have been summarized by 
Miss Elizabeth L. Wood of Clark University (43). Under the 
head of “Methods of Presentation and Learning” she concludes 
that the evident “want of agreement shown in the studies so far 
described” evinces the fact that the problem of whether an audi¬ 
tory, a visual, a visual-auditory, or a visual-auditory-motor pres¬ 
entation is “far from solution.” Both Pohlman (25) and Meu- 
mann (20) believe “that pupils should be taught to use all sorts 
of presentations though not forced into uneconomical methods”- 
(8). Von Sybel (40) in 1909 made experiments of similar im¬ 
port with 17 adults, students of rank; so also in 1912 Professor 
Henmon (9) of Colorado University reported experiments with 
six of his students from which he deduced the following interest¬ 
ing conclusions: 

“I. Auditory presentation is best in almost all cases. 

“II. Visual-auditory presentation is superior to visual alone in 
87 % of the cases. 

“III. Visual-auditory-motor is inferior to visual-auditory in 
59% of the cases; is inferior to auditory alone in 55% of the 
cases, to visual alone in only 15%. Therefore, a simultaneous 
appeal to several sense departments at once is of no advantage. 


35 


“IV. The relative value of thg different modes of presentation 
remains unchanged for one, two, or three presentations. 

“V. Individual differences in the amount retained was high.” 

Thorndyke (38), Kuhlman (12), Segal (30), and Weber (41) 
have also made interesting and important investigations along this 
line. Among these Kuhlman finds, in direct opposition to Hen- 
mon, that the poorest results come from auditory consciousness. 
Segal concluded that every individual should keep to his own type 
of imagery—rather a difficult pedagogical principle for a class 
room teacher with forty different individuals to teach to put into 
practice, yet doubtless of much significance after all, if every 
such teacher could ho an expert psychologist and investigator and 
be able to divide her class into sections, one containing only 
visuelles, another auditives, another motives, &c., according to the 
individual memory proclivities of the pupils. 

The results of Weber show that retention is in direct rela¬ 
tion to the number of repetitions, other things being equal. This 
is also in general accord with experiments in committing memory 
gems, made by the writer some years ago, the results of which 
are on file in the Department of Education of the University of 
Pennsylvania (10). Another conclusion of Weber, also espe¬ 
cially significant in our method of presenting this work, was that 
“it is economical to divide big tasks to be memorized into smaller 
ones, recognized as parts of the whole, but which can be learned 
without exhaustion.” 

In the light of these experiments, which unfortunately were 
carried on with groups too small to make their conclusions more 
than tentative, our method of presenting the combinations cannot 
be objected to on psychological grounds. 

Lest anyone should misunderstand this statement, however, 
and hold that we have violated both pedagogically and psychologi¬ 
cally the very principle of practice the necessity of which this 
study seeks to demonstrate in the very first sentence of our “Di¬ 
rections to Teachers for Conducting the Drills and Tests” 
(IX, 2), we must urge that the validity of our results depended 
upon the rigid adherence to the practice of equal daily emphasis, 
even though it did violate what we know perfectly well to be 
the best practice in teaching. Without this uniformity of em- 


phasis, it would have been impossible to judge relative difficulty 
by the results obtained. If just the right emphasis had been put 
upon each combination for its mastery, we should have expected 
just the same number of errors to appear in 4 X 3 as in 8 X 6, if 
errors appeared at all, and relative difficulty *would have been 
already worked out in practice, a situation which is contrary to 
fact. So much then had to be sacrificed for the sake of the experi¬ 
ment. 


XI. DIFFICULTY. 

1. ITS MEANING. 

Since this is a study concerning difficulty, it would seem ad¬ 
visable to discuss, at least briefly, some of the general charac¬ 
teristics of the term. It would also seem advisable to discuss it 
in the light of the various phases of our investigation. 

In teaching we impart knowledge; by stimulating continuous 
use of that knowledge we develop skill. The first three phases 
of our experiment, namely, th© determination of relative diffi¬ 
culty as shown in learning the facts, then as shown .by the num¬ 
ber of errors made, and finally as shown by the comparative num¬ 
ber of combinations forgotten, we are dealing for the most part 
with knowledge; in the last phase of our investigation, namely, 
the time tests, our principle interest is the measurement of skill. 
We are concerned, in the first place, with relative difficulty from 
the standpoint of acquisition and retention, and, in the second 
place, we note relative difficulty from the standpoint of use. 

Difficulty itself is a relative term. Existing in a small degree 
it represents facility, or ease of accomplishment. In its highest 
degree it requires the addition of a modifying word to convey 
its meaning. Then, too, what is difficult for one individual may 
be very easy for another. What is difficult for a given individual 
at a given time or under certain circumstances may be very easy 
at another time or under other circumstances. Any experiment, 
therefore, which seeks to determine relative difficulty must clearly 
set forth, as we have sought to do in this, the details indicated 
by the following questions: difficult for whom, at what age and 
grade, and under what method of instruction? 


37 


2. ITS CAUSE. 

As has already been pointed out, memory for facts is as¬ 
sisted by richness of association, use of judgment, &c. If there 
are combinations around which the associations are few or remote, 
as in the case of 8 + 5 = 13, or 9 X I = 63, where the figures 
in the answer are all different from those in the combination, as 
Professor Doring notes (2) ; if their nature is such as to make 
difficult the use of judgment, as in the case of 8 + 6 = 14, or 
8 X 12 = 96, where the very bigness of the results is on or be¬ 
yond the borderland of the comprehension of the pupil, we may 
expect difficulty in acquisition and in retention, and slowness and 
inaccuracy in use. If, on the other hand, the associations are many 
and easy, occurring in the child’s daily activities, as in the value 
of coins, in the purchase of rolls from the baker, or bananas, or 
pieces of candy from the shop-keeper, in the playing of marbles or 
dominoes, &c.; if the answers contain numbers which occur in 
the combinations, as I X 5 = 35, or 8 X 6 = 48; if the judg¬ 
ments are easy, as 5 + 5 = 10, then 5 + 6 = 11; if the answers 
are well within the experience or easy comprehension of the child 
—in all these cases we may expect little difficulty either in acquisi¬ 
tion or retention, and we may look for a high degree of skill and 
accuracy in the use of the combinations. 


3. ITS MEASURE. 

As, the acquisition of knowledge means, psychologically, the 
formation of new “brain paths,” as Professor James expresses it, 
difficulty might -be expressed in terms of neural resistance—in 
mental “ohms,” as it were. Aside from the physiological condi¬ 
tion of increased blood supply to the brain during the process of 
special activity of that organ, we do not know enough of the nature 
of its functioning to measure that functioning in comprehensible 
terms. We must content ourselves at present by trying to measure 
it in terms of its external phonomena, as we measure that mys¬ 
terious force called electricity. How do we determine that an 


38 


activity is difficult ? In the first place, other things being equal, 
by the time it takes to learn to perforin it with precision. Writ¬ 
ing is a difficult art for the child to acquire. Months, perhaps 
years, of practice are necessary to develop freedom and skill in its 
use. In the next place, we determine difficulty by the persistence 
of error in the course of mastery. How many false notes are 
struck in the process N of learning the scales; how many are made 
too long or too short, in learning to play a musical instrument? 
The smaller the chance for error, the less difficult the process. It 
is, therefore, less difficult to learn the piano than the pipe organ. 
One-step problems in arithmetic are less difficult than two or three- 
step problems. 

We should, therefore, evaluate our data in the first experiment 
by correlating the time required to reach the results obtained 
with the number of errors made in the process. In attempting to 
do this we found that the time given to each of the groups was not 
a true index of difficulty, because, for the sake of class efficiency, 
the drills had to be prolonged beyond the time necessary to secure 
efficiency with those who were not eliminated on account of absence 
from some necessary part of the work. The girls had sometimes 
to “mark time” while the boys were catching up. In the last ex¬ 
periment, however, as time was the only element considered, only 
correct results being counted, we may be justified in using it as a 
true index of difficulty as far as a comparison of the groups of 
facts indicated is concerned. 

The factor of time having, in a measure, been eliminated by 
force of circumstances in the evaluation of the results of the first 
experiment, w T e have laid down as fundamental the following two 
methods of evaluation: (1) In the process of learning the com¬ 
binations, the co-efficient of difficulty is the ratio of the number of 
errors made in the process to the number of errors overcome > as 
shown by a comparison of the number of e'nrors made in the pre¬ 
liminary test with the number of errors made in the final test in 
each combination f and reduced, for the purpose of comparison , to 
a basis of 50 or 100 pupils. In calculating co-efficients from this 
point of view, errors in the review tests were counted the same as 
errors in the course of instruction. (2) The coefficients of difficulty 
by the second method of evaluation are determined by dividing the 


39 


grand total of errors made in each combination by the number of 
pupils involved. This method gives as the co-efficient of difficulty 
the average number of errors for each pupil. The method is based 
upon the following reasoning: the presence of error in the pre¬ 
liminary test is an indication of difficulty—the difficulty to be 
overcome; the presence of error in the process of instruction is 
an indication of difficulty—the difficulty of mastery; the presence 
of error in the review and in the final test is an indication of diffi¬ 
culty—the difficulty of retention; hence the total of these errors 
would seem to indicate the entire difficulty presented by the com¬ 
bination. We have considered both of these methods of such value 
as to be worthy of record and have given the results of each side by 
side in Table VI. 

In the second and third experiments we have worked on the as¬ 
sumption that the number of errors made in the tests is in direct 
proportion to the difficulty, and have arranged the results in these 
experiments accordingly. Difficulty then is to be measured by the 
“method of errors” in the first three parts of our investigation, and 
by the “method of time” in the last part of the investigation. 


XII. RESULTS IN THE STUDY OF ADDITION. 

1. RESULTS IX TEACHING THE FORTY-EIVE ADDITION 
COMBINATIONS. 

To give in full the tabulation for all the classes for all the 
days for all of the combinations would require more space than 
is at the disposition of the writer. Those who may.be interested 
may have access to the complete records by applying to the De¬ 
partment of Education of the University of Pennsylvania. How¬ 
ever, in order to show the method pursued, the data tabulated 
for the study of 8 -f 3 is given below as typical. 


40 




73 

o 


EH 


•;uapgja-oo 

S.lO.T.I^l 



•S.TO.I.ig 
Ii?;ox pnc.io 



•sjo.ua ppx 

C H w >5 h C fO O 
<M rH H 

-H Cl I 

! 

Number of Errors. 

•}S9X 

HOCHTfrlCO 

1- 


•isax 

AVaiA3>X 

' OOrlOHCOOO 

o 


-tea l E0I 

; ; ; w ; ; ; 



’tea iR6 

: : : : : 

Cl 


-tea ms 

■ ) * ) ® * ) * 

o 


•^a qu 

• • O • T—1 • rH 

Cl 


■tea qi9 

CC • O • tH • tH • 

Cl 


-tea q;e 

O • O • CC tH o • 



-tea qif 

O • tH • H H (M • 

1C 


-tea ps 

o • • tH tH Cl • 



-tea Po 

HCCOHN5IC 

CC 


-tea m 

o c ci c h o: co o 



•4sax 

A.imiiiuip.Tx 

H ^ti H ^ ^ O 

tH 

GO 

Cl 


•aSy aSu.iaAy 

^ ~v?l vM v£N 

i-T' ri'' rO rO 

ChhOOOHt- 



•apu.xo 

p;<-^p;pq<ipQpq 
H H H Cl H H Cl Cl 



*s{idn ( | -o^ 

lO <M ^ b- (M CC Tti o 
tH t-i 

CO 

lO 


COMBINATION, 

8 + 3. 


• ? 
. K 

. C 

! c 

r* 

• r- 

m H- 

i 

c 

X ^ 

c r, 

4J If 

O £ 

H 

rv 

) 

; 

i 


co 


CO 

© 


©rHrH©^©©© 


OONNCOOrlO 


1C K C n C ri O LO 


Ol vM sN Ol 

rv" n' r4 Ss 

OCt*l'Ctthh 


ri H H M H H Ol IM 


l— l— 1.0 ttCKOQO 


© 


GO 


O Tfi M M t-d Cl © 
iH Cl 

CO o 
o 


i 

OOOCIOHOH 

l- • 

In 

©CClOOOrH© 

CO • 

CC 

. . ! . w . . . 

o • 

© 

: : : : c : : : 

o • 

Cl 

! ! ! 1 1-1 ! ! ! 

rH • 

rH 

0 

0 

0 

o • 

Cl 

CC • O • tH • CC • 

rH • 

CO 

O • rH -cicc • 

CO • 

*- 

rH • O' • Cl C C • 

CO • 

• 

CC 

© • Cl • o o o • 

• 

. 

Cl • 

• 

© 


a 
o 
© 

73 02 


05 „ 


02 


Eh h 


ci 


i'¬ 


ll- 

o 


CO 

o 












































































































































41 


The same kind of a study was made of each of the forty-five 
addition facts, an entire summary of which, tabulated to show 
co-efficients of difficulty from the two points of view as set forth 
on page 38, is shown below: 

TABLE IV.—PART I. 


SUMMARY OF RESULTS IN TEACHING ADDITION. 



Boys. 

Girls. 

1 

a 

b 

c 

d | 

i 

I 

e ! 

f 

g 

h 

a 1 

b 1 

1 

c 1 ! 

d 1 

e 1 

1 

f 1 | 

g 1 | 

h 1 

1 . 

57 

12 

41 

i 

1 

n 

3.73 

3.27 

0.95 

4S 

6 

34 

2 

4 

8.50 

8.85 

0.88 

9! 

57 

28 

59 

5 

23 

2.56 

2.25 

1.62 

48 

18 

45 

4 

14 

3.20 

3.33 

1.40 

3. 

57 

29 

63 

2 

27 

2.33 

2.04 

1.65 

48 

21 

39 

2 

19 

2.05 

2.14 

1.29 

4. 

57 

32 

55 

6 

26 

2 12 

1.86 

1.63 

48 

19 

41 

3 

16 

2.56 

2.66 

1.31 

5 + 1.... 

57 

29 

59 

3 

26 

2.27 

1.99 

1.60 

4S 

19 

43 

3 

16 

2.69 

2.80 

1.35 

6. 

58 

25 

57 

2 

23 

2.48 

2.10 

1.45 

55 

22 

22 

4 

18 

1.22 

1.11 

0.87 

7 . 

58 

27 

52 

2 

25 

2.08 

1.79 

1.40 

55 

19 

17 

5 

14 

1.21 

1.10 

0.75 

8. 

58 

26 

49 


24 

2.04 

1.7611.33 

55 

23 

19 

3 

20 

0.95 

0.86 

0.82 

9. 

58 

29 

57 

O 

o 

26 

2.19 

1.89 

1.53 

55 

23 

23 

5 

18 

1.28 

1.16 

0.93 

9 

5S 

22 

46 

2 

20 

2.30 

1.98 

1.22 

55 

20 

40 

3 

17 

2.35 

2.13 

1.15 

3. 

61 

28 

64 

4 

24 

2.66 

2.18 

1.57 

63 

31 

45 

3 

28 

1.61 

1.28 

1.25 

4. 

61 

32 

57 

5 

27 

2.11 

1.73 

1.54 

63 

30 

51 

4 

26 

1.96 

1.55 

1.35 

5. 

61 

23 

66 

3 

20 

3.30 

2.70 

1.51 

63 

2S 

54 

4 

24 

2.25 

1.78 

1.36 

6 + 2.... 

61 

21 

68 

4 

17 

4.00 

3.27 

1.53 

63 

34 

47 

5 

29 

1.62 

1.28 

1.36 

7. 

61 

38 

71 

3 

3o 

2.03 

1.66 

1.84 

63 

38 

49 

11 

27 

1.81 

1.43 

1.55 

8. 

57 

25 

37 

3 

22 

1.6S 

1.47 

1.14 

54 

15 

24 

5 

10 

2.40 

2.22 

0.S6 

9. 

57 

2S 

42 

7 

21 

2.0011.75 

1 

1.35 

54 

24 

26 

o 

o 

21 

1.24 

1.15 

0.98 

3. 

57 

25 

29 

4 

2l'il.3S!1.21 

1.02 

54 

23 

IS 

5 

IS 

1.00 

0.93 

0.85 

4. 

57 

25 

38 

8 

17 

2.24|1.96 

1.25 

54 

20 

44 

6 

14 

3.14 

2.91 

1.30 

5 . 

57 

28 

36 

6 

22 

1.6311.43 

1.23 

54 

22 

48 

8 

14 

3.43 

3.1ST.45 

6+3.... 

53 

20 

36 

6 

14 

2.57 

2.42 

1.17 

56 

32 

32 

9 

23 

1.3911.24 1.30 

7 . 

53 

29 

32 

4 

25 

1.2811.21 

1.23 

56 

27 

33 

8 

19 

1.74 

1.55 

1.21 

8. 

53 

28 

41 

7 

21 

1.9511.84 

1.43 

56 

29 

27 

7 

22 

1.23 

1.10 

1.13 

9 . 

53 

18 

43 

5 

13 

3.31 

3.12 

1.25 

56 

31 

3S 

7 

24 

1.58 

1.41 

1.36 

4 . 

53 

12 

16 

3 

9 

1.78|1.68 

0.59 

56 

• 20 

28 

o 

18 

1.55 

1.38 

0.89 

5. 

55 

27 

34 

4 

23 

1.48 1.35 

1.18 

55 

25 

43 

6 

19 

2.26 

2.06 

1.35 

6+4.... 

55 

30 

51 

3 

27 

1.89|1.72 

1.53 

55 

24 

41 

10 

14 

2.92 

2.55 

1.36 

7. 

55 

29 

41 

7 

22 

1.86|1.69 

1.40 

55 

23 

39 

5 

18 

2.16 

1.96 

1.22 

8. 

55 

29 

44 

9 

20 

2.20|2.00 

1.49 

55 

25 

35 

9 

16 

2.18 

1.98 

1.25 

9. 

55 

33 

50 

9 

24 

2.081.89 

1.67 

55 

34 

42 

8 

26 

1.61 

1.46 

1.53 

5. 

53 

18 

10 

2 

16 

0.6210.58 

0.57 

60 

16 

14 

3 

13 

1.08 

0.90 

0.55 

6. 

53 

24 

25 

6 

1811.39|1.31 

1.04 

60 

32 j 26 

9 

23 

1.13 

0.94 

1.12 

7 + 5.... 

53 

29 

36 

7 

22 

1.6311.53 

1.36 

60 

351 35 

11 

24 

1.46 

1.21 

1.35 

8. 

53 

22 

33 

9 

13 

2.54 

12.39)1.21 

60 

36 

| 36 

11 

25|1.44 

1.20 

1.38 

9. 

53 

27 

26| 11 

16jl.62ll.53 

11-21 

60 

37) 31 

S 

29fl. 07 j 0.89j 1.27 















































































1-2 


TABLE IV.—PART I.— Continued. 

SUMMARY OF RESULTS IN TEACHING ADDITION. 



Boys. 

Girls. 


a 

b 

c 

d 

e 

f 

g 

li 

a 1 

b 1 

c 1 

d 1 

e 1 

i 

1 f 1 

g 1 

1 h* 

6. 

54 

21 

18 

3 

IS 

1.00 

0.92 

0.79 

45 

20 

19 

5 

15 

1.27 

1.41 

0.98 

7. 

54 

21 

49 

8 

13 

3.77 

3.49 

1.45 

45 

27 

28 

8 

19 

1.47 

1.63 

1.40 

8+6.... 

54 

33 

55 

12 

21 

2.62 

2.42 

1.85 

45 

24 

35 

10 

14 

2.50 

2.77 

1.53 

9. 

54 

37 

66 

9 

28 

2.36)2.18 

2.04 

45 

28 

35 

9 

19 

1.84 

2.04 

1.60 

7. 

54 

30 

26 

6 

24 

1.0811.00 

1.15 

45 

22 

23 

4 

18 

1.28 

1.42 

1.09 

8 + 7.... 

66 

47 

76 

12 

35 

2.17 

1.64 

2.05 

57 

42 

67 

8 

34 

1.97 

1.70 

2.05 


66 

42 

96 

12 

30 

3.20 

2.42 

2.27 

57 

45 

75 

13 

32 

2.34 

2.10 

2.33 

S. 

66 

43 

47 

7 

36 

1.31 

0.99 

1.47 

57 

36 

27 

7 

29 

0.93 

0.81 

1.23 

9 + 8.... 

66 

43 

85 

15 

2S 

3.04 

2.30 

2.17 

57 

43 

53 

9 

34 

1.56 

1.37 

1.84 

9 + 9.... 

66 

42 

47 

7 

35 

1.34jl .01 

1.45 

57' 

38 

35 

3 

35 

1.00 

0.88 

1.33 


Note.— a and a 1 , number of boys and girls, respectively ; b and b 1 , number of er¬ 
rors made by boys and girls, respectively, in the preliminary test; c and c\ num¬ 
ber of errors made in learning ; d and d 1 , number of errors in final test; e and e 1 , 
(b — d) number of errors overcome; f and f 1 , co-efficients of difficulty (c -F e) ; g 
and g 1 , co-efficients of difficulty for 50 boys or 50 girls (50f -Fa); h and h\ total 
errors co-efficients of difficulty (b + c + d'l -Fa. 














































TABLE IV.—PART II. 

TOTALS FOR BOYS AND GIRLS. 


1 

■ A I 

B 

C 

E> 

E | 

F | 

G 

II 

1 . 

105 

IS 

75 

3 

15 

5.00 

4.76 

0.91 

•> 

105 

46 

104 

9 

37 

2.81 

2.68 

1.51 

3. 

105 

50 

102 

4 

46 

2.22 

2.11 

1.49 

■*. 

105 

51 

96 

9 

42 

2.28 

2.17 

1.4S 

5 + 1. 

105 

48 

102 

6 

42 

2.43 

2.31 

1.56 

G . 

113 

47 

79 

6 

41 

1.68 

1.49 

1.17 

7 . 

113 

46 

69 

7 

39 

1.77 

1.56 

1.08 

8. 

113 

49 

68 

5 

44 

1.55 

1.37 

1.08 

9 . 

113 

52 

SO 

8 

44 

1.S2 

1.61 

1.24 

o 

113 

42 

86 

5 

37 

* 2.32 

2.05 

1.18 

3. 

124 

59 

105 

7 

52 

2.02 

1.63 

1.41 

4 . 

124 

62 

108 

9 

63 

2.04 

1.65 

1.43 

5 + 2 . 

124 

51 

120 

7 

44 

2.53 

2.04 

1.43 

6 . 

124 

55 

135 

9 

46 

2.50 

2.02 

1.44 

7 . 

124 

76 

120 

14 

62 

1.94 

1.56 

1.69 

8 . 

111 

40 

61 

S 

32 

1.90 

1.73 

0.98 

9 . 

111 

52 

68 

10 

42 

1.62 

1.46 

1.17 

o 

O . 

111 

48 

47 

9 

39 

1.20 

1.08 

0.94 

4 . 

111 

45 

82 

14 

31 

2.64 

2.38 

1.27 

5 . 

111 

50 

84 

14 

36 

2.33 

2.10 

1.33 

G + 3 . 

109 

52 

68 

15 

37 

1.84 

1.69 

1.24 

7 . 

109 

56 

65 

12 

44 

1.48 

1.36 

1.22 

8 . 

109 

57 

6S 

14 

43 

1.5S 

1.45 

1.28 

9. 

109 

49 

81 

12 

37 

2.19 

2.01 

1.30 

4. 

109 

32 

44 

5 

27 

1.63 

1.50 

0.74 

5 . 

110 

52 

77 

10 

42 

1.83 

1.61 

1.26 

G + 4 . 

110 

54 

92 

13 

41 

2.24 

2.04 

1.45 

7 . 

110 

52 

80 

12 

40 

2.00 

1.82 

1.31 

8 . 

110 

54 

79 

18 

36 

2.19 

1.99 

1.37 

9 . 

110 

67 

92 

17 

50 

1.84 

1.67 

1.60 

5 . 

113 

34 

24 

5 

29 

0.83 

0.73 

0.56 

6 . 

113 

56 

51 

15 

41 

1.24 

1.10 

1.08 

7 + 5 . 

113 

64 

71 

18 

46 

1.54 

1.36 

1.35 

8. 

113 

58 

69 

20 

38 

1.81 

1.60 

1.30 

9. 

113 

64 

57 

19 

45 

1.27 

1.12 

1.24 

G. 

99 

41 

37 

S 

33 

1.12 

1.13 

0.87 

7. 

99 

4S 

77 

16 

32 

2.40 

2.42 

1.42 

8+6. 

99 

57 

90 

22 

35 

2.57 

2.60 

1.71 

9. 

99 

65 

101 

18 

47 

2.15 

2.17 

1.86 

7. 

99 

52 

49 

10 

42 

1.17 

1.18 

1.12 

8 + 7. 

123 

89 

143 

28 

69 

2.07 

1.68 

2.05 

9. 

123 

87 

171 

25 

62 

2.76 

2.24 

2.30 

8+8. 

123 

79 

74 

14 

65 

1.14 

0.93 

1.36 

9. 

123 

86 

138 

24 

62 

2.22 

1.80 

2.01 

9 + 9. 

123 

80 

82 

10 

70 

1.17 

0.95 

1.40 


Note. —For explanation see note at bottom of Table IV.—Part I. 






































































■ 1.4 















































































































































































































































































































































































































































































































































































































































































































































































































































































45 



-£at?aaa r jo aarSM?// 























































































































































































































































































































































































































































































































































































































































































































































46 


5. DISCUSSION" OF RESULTS. 

a. Interpretation of Table IV .—In the summary of Table III, 
shown in this table, it will be noted that the number of pupils 
varies for nearly every new group of facts presented. This is due 
to the elimination of pupils on account of absence. In order 
to be counted in the work a pupil must have taken the prelim¬ 
inary test, the review test, and the final test at the end of the 
term. Besides this he must not have been absent more than one 
day in every five days of intensive drill. As a result of these re¬ 
quirements at least 25% of the pupils who took part in the ex¬ 
periment had to be eliminated at one time or another in making 
up the results for the various groups. 

Owing to this varying number it became necessary to reduce 
results to a common basis to facilitate comparison. It would be 
all right to compare with each other the co-efficients of difficulty 
obtained for boys for the first five combinations, for in each 
case there were 57 boys involved, but to compare the same with 
group three where there are 61 boys is unfair, as is shown by 
the following figures: the co-efficient of 4 -j- 1 in column “f” 
is 2.12 for 57 boys, that of 4 + 2 in the same column is 2.11 for 
61 boys. A comparison here would indicate a difference of only 
.01, but when these figures are to be reduced to a basis of 50 boys 
each, as shown in column “g,” the first co-efficient becomes 1.86 
and the second 1.73, a difference of .13. The values in columns 
“f,” “f 1 ,” and “F” were at first reduced to a basis of one pupil. 
This, however, gave such a very small quantity that it was thought 
best to make the common basis a round number near in value to 
average number of pupils involved, so 50 was selected for the 
multiple of “f” and “f 1 ,” and this required the multiple of “V” 
to be 100. 

In this table the co-efficients shown in columns “g,” “g 1 ,” and 
“G” are those used for comparison for boys, girls, and the average 
for both boys and girls, while columns “h,” “h 1 ,” and “H” show 
in the same manner what we prefer to call the “Total Errors 
Co-efficient” for boys, girls, and the average for both boys and 
girls respectively. 


47 


b. Method of Studying the Graphs. —The values of the columns 
headed “g” and “g 1 ” (Table IV) are shown by the ordinate 
values of Graph I. In the study of the graphs in this part 
of the experiment we must always bear in mind that the co-effi¬ 
cients of difficulty found are co-efficients of difficulty in the process 
of learning the combinations in the order of their presentation. 
Let us take the three facts, 5 + 1? 1 + 3, and 8 + 4 for boys. 
It will appear that for each of these the co-efficient of difficulty 
is 2. We cannot reason from that that if these three facts were 
brought together they could be taught de novo with the same ease, 
but it does mean that the 5 + 1 fact at the time of its presenta¬ 
tion offered the same degree of difficulty as the 4 + 3 and the 
8 + 4 facts did at the time they were presented. If a line were 
drawn through the average heights of the ordinates in Graph I 
from left to right, its trend would be generally downward in¬ 
stead of upward as one might suppose. This demonstrates statis¬ 
tically in the study of numbers what everybody, even those who 
no longer believe in the doctrine of formal discipline, recognizes 
to be true, viz., that one of the results, one of the bi-products, of 
accomplishment in any field of endeavor is power —power to 
overcome with greater ease other difficulties in the same or in 
related fields of endeavor. 

c. Analysis and Significance of the Graphs. —A further analysis 
of Graph I shows that in 14 out of the 45 cases the “curves” for 
boys and girls tend in opposite directions as in the case of 5 + 3, 
for instance. In one case, 8 + 5, the “curve” for girls is horizon¬ 
tal while for boys it tends strongly upward. In 30 cases, there¬ 
fore, while the values for boys and girls differ materially, the 
tendencies are in the same direction as compared with the co¬ 
efficients of the combinations immediately preceding. The aver¬ 
age of the co-efficients for boys is 1.89 while that for the girls is 
1.82. This would indicate that the girls learned the combina¬ 
tions with slightly less difficulty than the boys, and this notwith¬ 
standing their inordinately large co-efficient for learning 1 + 1. 
Had the co-efficient for 1 + 1 been the same for both boys and 
girls, the difference between the averages of their respective co¬ 
efficients of difficulty would have been .18 instead of .07, or would 


have shown that girls learn the combinations with abont 10% 
less difficulty than boys of the same age and grade. 

The high co-efficient of difficulty for 1 —)— 1 for the girls is due 
to the fact that only six errors were made by the 48 girls in 
this combination in the preliminary tests, the possibilities for im¬ 
provement were, therefore, correspondingly less. If no errors, 
instead of two, had occurred in the final test, the co-efficient 
would then have been 5.91. Both of these figures demonstrate 
the necessity of studying the co-efficients here represented in 
connection with those shown in Graph II. 

A glance at Graph II, which shows the total errors co-effi¬ 
cients of difficulty, calculated on the basis of the entire number 
of errors made by boys and girls in all of the tests, shows that the 
girls have a smaller average co-efficient than the boys, the aver¬ 
age in this case being 1.41 for boys and 1.26 for girls, or about 
10% less. This graph shows a very close similarity between the 
“curve” for boys and that for girls, the fewer errors appearing 
in the double numbers being especially significant for both. 

We have no doubt that a part of the difficulty represented in the 
first group was due to the mechanical difficulty of writing the 
numbers required by the tests, though, as has already been ex¬ 
plained, most of the 1—B children had had the writing of num¬ 
bers in their work in the reception grade. This difficulty, how¬ 
ever, would be at its maximum in September and applies to only 
about one-fourth of the pupils whose records are given. 

In order to obtain the best results in the study of Graphs I 
and II, it is best to study them by groups of five combinations 
each, that is, in the manner in which the work was given. It 
is readily seen that the learning of the first group (1 + 1 to 
1 + 5) carried over to make the second group (6 + 1 to 9 + 1) 
easier; but the fact 2 + 2 of this group represented a new order 
of things, consequently a new difficulty. With these graphs be¬ 
fore her and an occasional check upon the results of her teach¬ 
ing, the primary teacher should be able to attack with some con¬ 
fidence the teaching of the elementary combinations in addition. 


49 


XIII. RESULTS IN THE STUDY OF MULTIPLICATION. 

1. RESULTS IN TEACHING THE SEVENTY-EIGHT MULTIPLICATION 
COMBINATIONS. 

For the sake of brevity, as in the case of the results of the 
study of addition, we give the detail of but one (8 X of the 
seventy-eight combinations studied. This will serve to demon¬ 
strate the method of tabulating the results for all. The complete 
record may be had by applying to the Department of Education 
of the University of Pennsylvania. 

4 


TABLE V. 

BOYS. 


50 


•}UdIDl{Jd-Of) 

sao.ug 

::::::::: 


•sjo.u^r 


240 

•S.IOJ.ia tBIOX 

GO © L- © CO © 
H Tft r-i (M rH tH CO CO 

BO BO 
© CO 

01 

Number of Errors. 

[«nij 

Cl ^0 H CO H H l'- © (X 1 

l- 

co 


AV9IA0JJ 

•A'uq MU 

HHHrtCdCCO 

rH rH 

01 

fY** 

• a' 


••••••••a 

••••••••• 

••••••••a 

•••••aaaa 

• 

• 


*^«a Mb) 

• • • • C ^ a a a a 

•••• aaaa 

••aa a a a ' a 

••aa aaaa 



•^a mis 

^ • • • • 

• • • • • 

• • • • • • 

• • • # • 

rH 


*^a mif 

01 • © © © TH • • a 

# • • # 

# • • • 

• • • • 

CO 


*^«a ps 

H Cl C LO Cl rH H 

• • 

a a 

• • 

12 


m **a. Po 

H H ^ Cl C) H • rF 

. 

• 

t- 

tH 

| 

•A-u(i in 

01 CO rH © © 01 01 © © 

Ol 

01 

| 


BO BO t- © rH X BO CO BO- 1 

rH rH rH 

‘ 

rH 

X 


*oSy oSiuoAy 

CD cd r D «—». 

Wa w« Wa 1 W* Wa> Wa 

• 

• 

• 

• 

• 



01 01 01 CO CO CO CO ~~ O - ' 

• 

• 

• 

• 

• 


•siTdnj ;ox 

BO LO O rn H H CO CO X 1 i 

H H H H H tH 

X 


fc 

o 

w 

H 

< 

fc X 

HH 

C3 CC 1 

WH 

o 

o 

E 

i ora is . 

Errors second time over.1 



j 

a a a aa a -a a 

1 : : :. 

• • H 

CO 

! • 
01 









cc 

CC f C C C Lt T O 

r- X 


• 

CO Ol rH rH 01 Ol 

CO 01 


• 


rH 


• 

- 



• 

01 01 CO 01 rH CO 01 rH 

© • 



rH • 

BO 

©© rH © tH'BO © © 

Ol • 

jrH 

a 

Ol • 

• 

• 

• 

BO 

• •aaaaaa 

• • 

• 

aaaaaaaa 

• • 

• 

• •aaaaaa 


• 

• ••••••a 

• • 

a 

• • • C~; a a a a 

• • • • • | | 

• 

• 

• 

i O 

• aaaa 

• a a a • 

• aaaa 

• 

• 

• 

rH 

© • 01 rH © • • • 

• a a a 

• a a • 

• a a # 

00 • 

• 

• 

• 

© 

© rH rH © rH © 

CO • 

BO 

• a 

• a 

• a 

* 

rH 

H n n C CC r • © 

L- . 

1- 

# 

• 

• 

• 

Ol 

| r-( LC CC WHTfC Cl 

© • 

rH 


rn 

# 

a 

rf 

rf cc cs rr cc ci ci t-i 

r • 

01 

rH th rH 

« A 

w • 

rf 


a 

• 

rn 

H N 

* * 

• • 

• 

• 

X X X © ©a © © © 

, 

a a 

a a 

• 

• 

• 


• • 

• 

01 01 01 CO CO CO CO CO 

• • 

• • 

• • 

• • 

• 

• 

• 

• 

^ c c. rr x h m o: 


01 

r- 1-H rH 1 

1- • 



• 

• 

• 

rH 


a a 

a a 

• a 

a a 

a a 

• a 

a a 

• a 

• Z-4 

• 

• 

• 

* 

• 

• 

• 

• 

• 


a O' 

• 


a > 

• 


a W 




CC 


a C 



r* 
a £« 

C3 

4-> 


. H-> 

O 


: 

H 





r-* 

a 

© 



ioj 


V ™ 



C3 



4 i CC 



H 2 



?■* 



rvi 





























































































































































































51 


From the full records, of which Table V is a sample, the data as 
shown in Table VI below were obtained. This table represents 
the tabulation of nearly 13,000 errors, and is worked out on ex¬ 
actly the same principles as Table IV in addition, and,* therefore, 
needs no special comment as to its meaning. 


TABLE VI.—PART I. 

SUMMARY OF RESULTS IN TEACHING MULTIPLICATION. 


|] 

Boys. 



a 

b 

c 

d 

e 

f 

g | 

h ! 

3*. 

81 

21 

8 

12 

9 

0.891.10 

0.51 

6 . 

108 

46 

55 

20 

26 

2.12 

1.96 

1.12 

8x3.... 

108 

45 

73 

29 

16 

4.56(4.22 

1.36 

9. 

97 

35 

54 

25 

10 

5.40|5.56 

1.18 

11. 

97 

28 

23 

7 

21 

1.08|1.12 

0.60 

12. 

1 

97 

32 

50 

21 

J 

11 

4.55(4.69 

1.06 

| | 

4. 

97 

1 47 

23 

181 

1 29 0.79 

0.81 

0.91|| 

7 . 

95 

1 49 

91 

26 

23 3.96 

4.16 

1.751 

8.. 

95 

j 52 1 126 

28 ! 

| 24 5.25 

5.52 

2.17 | 

9x4.... 

95 

1 621130 

38 

245.42 

5.70 

2.42| 

10. 

99 

1 29 

15 

5 

| 24J0.62 

0.63 

0.50 

12. 

99 

1 

55 

54 

27 

1 

28|1.93 

i i 

1.95 

i 

1.37 

5 . 

99 

1 33 

14 

10 

! 2310.6110.62 

0.58 

7x5. ... 

101 

46 

51 

281 1812.831 

2.80 

1.24 

10 . 

101 

34 

14 

1 3 

i 31j0.45j0.45 

0.50| 

12 . 

95 

60 

31 

23 

37|0.84|0.88 

i 

1.20 

.i 

6 . 

95 

50 

30 

13 

37 0.8li0.85 0.98i| 

7 . 

| 95 

68 

101 

1 23 

4512.471 

2.60 

2.13 j 

8 . 

95 

66 

77 

98 

39 

27 3.63 3.8212.141 

9X6.... 

1 95 

(111 

j 43 

34j3.26j3.43 

|2.44| 

10 .1 

1 98 l 

43 1 

10 

1 8 

35|0.29|0.30|0.62| 

11 . 

98 1 

54 1 

23 

1 9 

48|0.48|0.49|0.85j 

12 . 

1 98 l 

| 72! 

| 67 

1 25 

47|1.43(1 - 46 

j 

1.67 

7 . 

1 98 

74 

| 37 

1 I 4 

60|0.62|0.63 

1.28 

8.| 

98 l 

811112 

| 37 

44 

2.77| 

2.82 

2.45 

9 . 

881 

841184 

43 

41|4.49|5.10 

3.531 

10X7. . .| 

' 881 471 

16 

5 

42 0.38|0.43 

0.77 j 

11.| 

881 

521 

31 

8 

44|0.71|0.80 

1.03! 

12.| 

| 881 65|102 

33 

32|3.19|3.62 

2.28| 


Girls. 


a 1 

b 1 

c 1 

d 1 

e 1 

1 

fl 1 


h 1 

60 

18 

8 

4 

14 

0.57|0.95 

0.50 

75 

29 

30 

8 

21 

1.431.90 

0.89 

75 

39 

34 

10 

29 

1.52 

2.02 

1.24 

70 

30 

48 

9 

21 

2.33 

3.33 

1.24 

70 

22 

19 

5 

17 

1.12 

1.60 

1.66 

70 

1 

31 

37 

8 

23 

1.61 

2.30 

1.09 

70 

34 

19 

4 

30 

0.63 

0.90 

0.77 

64 

40 

46 

7 

33 

1.39 

2.17 

1.45 

64 

33 

58 

9 

24 

2.42 

3.78 

1.56 

64 

41 

67 

11 

30 

2^23 

3.48 

1.86 

76 

30 

11 

1 4 

26 

0.42 

0.50 

0.59 

76 

1 

46 

| 37 

| 

10 

36 

1.03 

[1.35 

1.22 

76 

38 

14 

4 

34 

0.41 

0.54 

0.74 

72 

48 

31 

7 

41 

0.76 

1.05 

1.19 

72 

33 

14 

1 

32 

0.44 

0.61 

0.67 

77 

1 

46 

19 

9 

37 

0.51 

0.61 

0.96 

77| 

451 

19 

20 

25 

0.76 

0.99 

0.92 

77 

55 

1 69 

1 12 

43 

1.60 

2.0811.77 

77 

60 

1 75! 

1 14 

46|1.63 

2.12|1.93 

77| 

57| 

1 91 

j 20 

37 2.54 

3.30|2.22 

74| 

38 

1 4 

1 9 

35|0.11 

0.15|0.47 

74| 

42! 

| 13' 

1 4 

38|0.34 

0.46'O.SO 

74 

1 

58 

46 

1 io 

1 

48 

0.96 

1.30 

1.54 

74 

47 

21 

I 9 

38 

0.55 

0.74 

1.09 

74| 

61 

82 

1 16 

' 45 

1.82 

2.46 

2.15 

69| 

! 61 

1 76 

1 21 

40 

1.9012.75 

2.29 

69 

38 

8 

1 1 

37 

0.2210.32 

0.6S 

691 

1 36 

1 12 

I 4 

32|0.37|0.54 

0.75 

69| 541 55 

l I 9 

14|1.25|1.81 

1.72 


* Thirtv-four of the smaller combinations have been omitted in this table for 
typographical reasons, but their several co-efficients are included in the graphs 
involved. 
































































52 


TABLE VI.—PART I.— Continued. 

SUMMARY OF RESULTS IN TEACHING MULTIPLICATION. 


Boys. 


Girls. 


i 



a 

b 

c 

d 

e 


f 


or 

o 

1 

h 

a 1 

b 1 

1 c 1 

1 

1 d ' 

e 1 

f 1 

g 1 

h 1 

8 . 

88 

63 

110 

25 

38 

2 

.90 

3 

.29 

| 

3.25 

69 

48 

37 

10 

38 

0.97 

1.41 

1.38 

9 . 

102 

87 

94 

33 

54 

1 

.74 

1 

.71 

2.10 

75 

67 

40 

12 

55 

0.73 

0.97 

1.59 

10X8 ... 

102 

53 

16 

14 

39 

0 

.41 

0 

.40 

0.81 

75 

35 

5 

2 

33 

0.15 

0.20 

0.56 

11 . 

102 

56 

21 

5 

51 

0 

.41 

0 

.40 

0.80 

75 

45 

4 

4 

41 

0.10 

0.13 

0.71 

12 . 

102 

81 

60 

23 

58 

1 

.03 

1 

.01 

1.61 

75 

64 

46 

10 

54 

0.85 

1.13 

1.60 

9 . 

102 

70 

46 

13 

57 

0 

81 

0 

.79 

1.26 

75 

53 

16 

4 

49 

0.33 

0.44 

0.97 

10 . 

106 

58 

19 

14 

44 

0 

43 

0 

41 

0.86 

74 

33 

7 

3 

30 

0.23 

0.31 

0.58 

11X9 ... 

106 

58 

24 

10 

48 

0 

50 

0 

.47 

0.87 

74 

41 

14 

5 

36 

0.39 

0.53 

0.81 

12 ...... 

106 

89 

58 

30 

59 

0 

97 

0 

91 

1.67 

74 

62 

32 

10 

52 

0.62 

0.84 

1.41 

10 . 

106 

49 

17 | 

5 

44 

0 

39 

0 

37 

0.67 

74 

38 

7 

5 

33 

0.21 

0.28 

0.68 

11x10.. 

106 

85 

571 

24 

61 

0 

93 

0 

88 

1.57 

74 

63 

25 

10 

53 

0.47 

0.63 

1.33 

12 . 

100 

75 

58 

25 

50 

1 

16 

1 

16 

1.68 

74 

61 

40 

11 

50 

0.80 

1.08 

1.51 

11X11.. 

100 

89 

74 

20 

69 

1 

07 

1 

07 

1.83 

74 

60 

38 

8 

52 

0.73 

0.99 

1.43 

12 .| 100 | 

93 

951 

24 

69 

1. 

38 

1. 

38 

2.12 

74 

65 

59 

9 

56 

1.05 

1.42 

1.80 

12x 12..|100 

79 

45 

13 

56 

0 

80 0 

80 

1.37 

| 74 

59 

17 

8 

51 

0.33 

0.45 

1.14 


Note. —For explanation of columns, see note at end of Table IV., but here g and 
g 1 = 100f or lOOf 1 - 4 - a or a 1 , respectively. 











































TABLE VI.—PART II. 

TOTALS FOR BOYS AND GIRLS. 



A 

B 

C 

3*. 

141 

39 

16 

6. 

183 

75 

85 

8X3. 

183 

84 

117 

9. 

167 

65 

102 

11. 

167 

50 

42 

12. 

167 

63 

87 

4. 

167 

81 

42 

7. 

159 

99 

137 

8x4. 

159 

85 

184 

9. 

159 

103 

197 

10. 

175 

59 

26 

12. 

175 

101 

91 

5. 

175 

71 

28 

7X5. 

173 

94 

82 

10. 

173 

67 

28 

12. 

172 

106 

50 

6. 

172 

95 

49 

7. 

172 

123 

180 

8X6. 

172 

126 

173 

9. 

172 

134 

205 

10. 

172 

81 

14 

11. 

172 

96 

36 

12. 

172 

130 

113 

7. 

172 

121 

58 

8. 

172 

142 

204 | 

9X7. 

157 

145 

260 1 

10. 

157 

85 

24 | 

11. 

157 

88 

43 | 

12 .. 

157 

119 

157 | 

8. 

157 

111 

147 

9. 

177 

154 

134 

10X8. 

177 

88 

21 

11. 

177 

101 

25 

12. 

177 

145 

106 

9. 

177 

123 

62 

10. 

180 

91 ‘ 

26 

11X9. 

180 

99 

38 

12. 

180 

151 

90 

10. 

180 

87 

24 

11X10 .... 

180 

148 

82 | 

12. 

174 

136 

98 


D 

E 

F 

G 

H 

16 

23 

0.70 

0.98 

0.50 

28 

47 

1.81 

2.56 

1.03 

39 

45 

2.60 

2.84 

1.31 

34 

31 

3.29 

3.94 

1.20 

12 

38 

1.10 

1.32 

0.62 

29 

34 

2.56 

3.07 

1.07 

22 

59 

0.71 

0.85 

0.85 

33 

56 

2.45 

3.08 

1 63 

37 

48 

3.83 

4.81 

1.92 

49 

54 

3.65 

4.58 

2.20 

9 

50 

0.52 

0.59 

0.54 

37 

64 

1.42 

1.62 

1.31 

14 

57 

0.49 

0.56 

0.65 

35 

59 

1.39 

1.61 

1.22 

4 

63 

0.44 

0.51 

0.57 

32 

74 

0.68 

0.78 

1.09 

33 

62 

0.79 

0.92 

0.95 

35 

88 

2.05 

2.38 

1.96 

53 

73 

2.37 

2.76 

2.02 

63 

71 

2.89 

3.36 

2.34 

11 

70 

0.20 

0.23 

0.56 

10 

86 

0.42 

0.49 

0.825 

35 

*95 

1.19 

1.38 

1.62 

23 

98 

0.59 

0.79 

1.17 

53 

89 

2.29 

2.66 

2.32 

64 

81 

3.21 

4.09 

2.99 

6 

79 

0.30 

0.38 

0.73 

12 

76 

0.57 

0.73 

0.905 

43 

76 

2.07 

2.64 

2.03 

35 

76 

1.93 

2.46 

1.87 

45 

109 

1.23 

1.39 

1.88 

16 

72 

0.29 

0.33 

0.71 

9 

91 

0.27 

0.31 

0.76 

33 

112 

0.95 

1.07 

1.60 

17 

160 

0.59 

0.69 

1.13 

17 

74 

0.35 

0.39 

0.74 

15 

84 

0.45 

0.50 

0.84 

40 

111 

0.81 

0.90 

! 1.56 

10 

77 

0.31 

0.34 

0.67 

34 

14 

0.57 

0.63 

| 1.47 

36 

100 

0.98 

1.11 

| 1.55 





































































TABLE VI—PART II.—Continued. 

TOTALS FOR BOYS AA'J) GIRLS. 



A 

B 

C 

D 

E 

F 

G 

H 

11x11.... 

174 

149 

112 

28 

121 

0.93 

1.07 

1.66 

12 . 

174 

158 

154 

33 

125 

1.24 

1.42 

1.98 

12 X 12 .... 

174 

138 

1- 

62 

21 

107 

0.58 

0.67 

1.27 


Note. —A, number of pupils ; B, number of errors in preliminary tests ; C, num¬ 
ber of errors made in process of learning: D, number of errors in final tests ; E 
(B — D), number of errors overcome; F (Ch-E), co-efficients of difficulty for 
group tested; G (200F -F A), average co-efficients of difficulty. See g and g 1 of this 
(B+C+D) 

table; H |- - - >, total errors co-efficients of difficulty. 


















54—a 
































































































































































































































































































































































































































































































































































































































































































































54—b. 



iiuuiuufuu. 


.L i L 4- 


















































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































55 


5. DISCUSSION OF THE RESULTS IN MULTIPLICATION. 

Hie number, age, sex, and class distribution of the pupils as 
shown in Table V represent the conditions in regard to these 
factors which obtained throughout the entire study. The experi¬ 
ment ran through two terms of five months each, beginning in Feb¬ 
ruary and ending the following February. Hence, the conditions 
of the whole school year, such as vacations, holidays, promotions, 
&c., are all involved and argue for the validity of the results under 
circumstances that are practically duplicated from year to year in 
any small system of schools. 

The variation in the number of pupils recorded in each group of 
combinations studied is due to the same process of elimination for 
absence as was described in the discussion of the addition results 
(XII 5, a). 

All results in multiplication are reduced to the same base to 
facilitate comparison, and the same method of evaluation, or de¬ 
termining co-efficients of difficulty used in addition is again used 
for the multiplication (XII 5, a). 

It must be borne in mind, as in the case of the addition graphs, 
that the results as shown by Graphs III and IV are of value only 
when the combinations are studied in the order of the group pre¬ 
sentations—that is, progressively from group one to group sixteen. 
The order of the combinations in each group was always pro¬ 
miscuous. By this method of study it will be noted how, for in¬ 
stance, the effect of learning group one (1 X 1 to 5 X 1) is shown 
in learning the other combinations with 1, and how the general 
trend of difficulty at first upward soon begins to go downward and 
so continues as the size of the numbers increase. The “curves” 
show very clearly the peaks of difficulty and the depressions indica¬ 
tive of the lack of difficulty, in fact, a study of the graphs tells the 
whole story. 

A further analysis of Graph III shows that in 15 out of the 78 
cases the lines of the “curve” tend in opposite directions for boys 
and girls. Therefore, in 63 cases the trend of difficulty for boys 
and for girls is in the same direction, if not to the same values. A 
casual glance at this graph will show that the girls had less diffi- 


56 


culty in learning the multiplication combinations than the boys. 
If we calculate the averages of their respective co-efficients of diffi¬ 
culty, we fine that the average for boys is 1.61, while that for girls 
is only 1.18, or nearly 27% less. In other words, the girls learned 
the multiplication facts more readily than the boys by over a 
fourth. 

Graph IV, showing the total errors co-efficients for boys and 
girls, presents a very interesting similarity of tendency for the two 
sexes. In this all the errors made in the course of the experiment 
are included. Here, again, we note an advantage in favor of the 
girls. A calculation of the average of the co-efficients of difficulty 
from columns in Table VI marked “h” and “h 1 ” gives for the 
boys 1.035, and for the girls .91. This shows that the girls made 
about 13% fewer errors throughout the drills than the boys. 

These multiplication, graphs should be of much assistance to the 
primary teacher who is confronted with the problem of teaching 
the multiplication facts. Aside from what they show of relative 
difficulty on their face, they should warn the teacher against an 
attempt to get along too rapidly in the early stages of the process. 
She should remember that here is the period for the development 
of power which is to serve her pupils well in their struggles with 
the larger numbers to come. 


XIV. RELATIVE DIFFICULTY AS INDICATED BY 
ERRORS ALONE. 

1. PLAN. 

It was not deemed sufficient to study this problem from the two 
points of view only that have thus far been set forth. These have 
their value to the teacher in the process of teaching the combina¬ 
tions. It was also thought advisable to consider the problem that 
is presented to the teacher after the ground has been gone over, to 
point out to her the places where the tares of forgetfulness are most 
likely to have sprung up and to have choked out the newly planted 
memories. Through the kindness and co-operation of the Super¬ 
intendent, Principals, and Third and Fourth Grade Teachers of 


57 


one of the large city systems of schools in New Jersey, the writer 
was able to get results from the examination of a much larger 
number of pupils than the foregoing experiments involved. 

The plan was to give each third grade pupil in the city a written 
test, first, in the forty-five addition facts, and immediately there¬ 
after, a test in the seventy-eight multiplication combinations. 
These tests were given during the last week of school preceding the 
summer vacation. It should be added here that the course of study 
in arithmetic provided for the teaching of all the tables to 12 X 12 
by the end of the third year. The plan also proposed exactly the 
same tests to be given to the pupils of the fourth grade during the 
first week of school in September. This grade was, of course, se¬ 
lected with the idea of catching a maximum number of those who 
were tested in June and comparing .the results of the two tests. 
This comparison would show the combinations most likely to be 
forgotten and thus provide the fourth grade teacher with a most 
valuable guide in directing the work of her number reviews. 


2. NATURE OF THE TESTS. 

With these ends in view tests in the forty-five facts of addition, 
promiscuously arranged, were given to 1,056 pupils, as indicated 
in Table VII, which shows the number of errors in the various 
combinations and the percentage of such errors. Table VIII 
shows the arrangement of these facts in the order of their difficulty, 
based upon the number of errors made in each. 

The seventy-eight multiplication combinations, arranged pro¬ 
miscuously in exactly the same way as for the preliminary and 
final tests given in the first part of this investigation, were given 
to 1,215 pupils, as indicated in Tables IX and X. 

The combinations, both in addition and multiplication, were 
typewritten and mimeographed. The directions to teachers which 
follow will explain more fully the nature of the tests, as well as 
explain the methods pursued in giving them. 


58 


3 . DIRECTIONS FOR TEACHERS. 


1. See that each pupil lias a usable lead pencil. 

2. To explain what is desired to your class : Take the addition blanks and 
hold them up before your pupils and say, “I have here some little examples in 
addition, and I want to see how quickly this class can write the answers after 
the — signs.” (Show the method on one of the papers and on the board, as 
12 + 12 = 24.) “You will each be given a sheet face down upon your desk. 
You are to fill in the blanks on the back of this first. (Show the place.) At 
the signal, ‘turn papers,’ you are to turn your sheet and write the answers, 
beginning with the left-hand column, and going down each in order. (Show 
how.) While you are doing this and at the end of every half minute, I shall 
say ‘mark.’ At that signal you will draw a line under the problem last fin¬ 
ished. (Show how on the sheet held before the class.) Is there anyone who 
does not understand?” If all understand, 

3. Distribute the papers as explained. 

4. Have each pupil fill in the blanks on each sheet (name, grade, date). 

When all have finished this, say, .“turn papers and write answers.” You will 
then say “mark” at the end of each half minute. 

5. As soon as each pupil has written all of the answers, he is to turn his 

sheet face down upon his desk and wait until the others have finished. Time 
should be given for all pupils to finish, but they are not to be told so. The 
maximum time given to the room for the exercise will be placed on the outside 
of each envelope. 

6. The same directions should precede the giving of the work in multiplica¬ 
tion, but the teacher will illustrate with 12 X 35 = 180. 

7. Pupils must be instructed not to “count up” in addition, and not to “run 
down the row” in multiplication. They are to write the answers as fast as 
possible and from memory. 

8. No explanations or interruptions are to take place after an exercise has 
begun. 

0. When the tests have been finished, place the papers in the envelopes in 
which they are given you and return the same to the Principal. 

10. As is indicated above, the addition test is to be given first. This is to 
be followed at once by the multiplication test. 


59 


TABLE VII. 


SHOWING THE NUMBER OF ERRORS MADE IN THE ADDITION COMBINATIONS BY 

1,056 CHILDREN. 



Boys. j 



Girls. 


Grand 

Totals. 

Per Cent. 

of Errors. 

Grade . 

3 

4 

1 & 2 

1 

Totals. 

3 

i 

4 

1 & 2 

Totals. 

No. Pupils... 

332 

115 

55 

502 

33S 

i 

112 

54 

554 

1056 


1 . 

4 

1 

i 

6 

8 

2 

1 

11 

17 

1.6 

2 

12 

• • • • 

2 

14 

6 

• • • • 

1 

7 

21 

2.0 

3. 

S 

• • • • 

1 

9 

6 

2 

2 

10 

19 

1.8 

4 . 

3 

1 

2 

6 

7 

1 

1 

9 

15 

1.4 

5 + 1. 

5 


2J 

7 

7 

1 


8 

15 

1.4 

6 .. 

6 


2 

8 

6 

3 


9 

17 

1.6 

7 . 

5 


3 

8 

8 

3 

1 

12 

20 

1.9 

8. 

4 


8 

12 

7 



7 

19 

1.8 

9 . 

3 

.... 

4 

7 

2 

3 

1 

6 

13 

1.2 

2 . 

3 

1 

1 

5 

4 



4 

9 

0.8 

3 . 

5 

1 

3 

9 

6 

1 

1 

8 

17 

1.6 

4 . 

7 


Q 

O 

10 

3 


3 

6 

16 

1.5 

5 + 2 . 

5 

1 


6 

4 


3 

7 

13 

1.2 

6 .. 

3 


4 

7 

8 

i 

3 

12 

19 

1.8 

7 . 

6 

1 

2 

9 

9 

3 

3 

15 

24 

2.3 

8. 

5 

2 

1 

8 

3 

• • • • 

1 

9 

12 

1.1 

9. 

2 

.... 

1 

3 

6 

3 

3 

12 

15 

1.4 

3 

2 

9 

1 

5 

2 

1 


3 

8 

0.7 

4. 

5 

9 

2 

9 

8 


1 

9 

18 

1.7 

5. 

5 

4 

3 

12 

7 

i 

6 

14 

26 

2.5 

6+3 . 

3 

2 

2 

7 

5 

2 

6 

13 

20 

1.9 

7 . 

8 

3 

6 

17 

1 10 

4 

6 

20 

37 

3.5 

8 . 

10 

4 

8 

22 

7 

7 

5 

19 

41 

3.9 

9 . 

14 

O 

o 

4 

21 

14 

5 

3 

22 

43 

4.1 

4 

3 

3 


6 



2 

2 

8 

0.7 

5 . 

5 

1 

3 

9 

S 

1 

2 

11 

20 

1.9 

6 + 4 . 

8 

3 

4 

15 

1 10 

5 

4 

19 

34 

3.2 

7. 

14 

3 

5 

22 

19 

4 

3 

26 

48 

4.5 

8. 

13 

6 

6 

25 

4 

5 

3 

12 

37 

3.5 

9. 

14 

5 

5 

24 

17 

5 

5 

27 

51 

4.8 

5 . 

4 

1 

. 

5 

1 


3 

4 

9 

0.8 

6. 

10 

5 

3 

IS 

7 

i 

7 

14 

32 

3.0 

7 + 5. 

10 

11 

9 

30 

11 

6 

8 

26 

56 

5.3 

8. 

19 

11 

5 

35 j 

23 

4 

6 

331 

68 

6.4 

9 . 

15 

4 

7 

26| 

13 

7 

3 

23 1 

49 

4.6 

6 . 

10 

1 

1 

12 

5 

1 

2 

8 

20 

1.9 

7 + 6. 

13 

6 

5 

24 

16 

4 

6 

26 

50 

4.7 

8 . 

17 

• 8 

11 

36 

14 

9 

7 

30 

66 

6.2 

9. 

16 

8 

rr 

i 

31 

[ 22 

14 

5 

51 

82 

7.8 













































































































60 


TABLE VII.— Continued. 

SHOWING THE NUMBER OF ERRORS MADE IN THE ADDITION COMBINATIONS BY 

1,056 CHILDREN. 



Boys. 

Girls. 

ns co 

-M K 

o> ? 

O u 

Grade . 

3 

4 

1 & 2 

Totals. 

3 

4 

1 & 2 

j Totals. 

03 -M 
f- O 

OH 

nH* 

CJ 

Cm o 


1 

I 



1 | 


1 


| 


| 

No. Pupils. .. 

332 

lloj 

55 

| 

502 

338 

| 

112 

1 

54 

554* 

1056 


7. 

8 

2 

4 

14 

5 

1 


6 

20 

1.9 

8 + 7. 

18 

8 

8 

34 

19 

7 

9 

35 

69 

6.5 

9. 

29 

8 ! 

8 

45 

27 

8 

10 

45] 

90 

8.5 

8+8. 

13 

3 

4 

20 

1 12 

2 

3 

17 

37 

3.5 

9. 

30 

12 

14 

561 

| 23 

5 

11 

39 

95 

9.0 

9 + 9. 

5 

4 

6 

15 

1 

j 7 

2 

3 

14 

29 

2.7 


The average number of errors made by each boy was 1.45; that 
for each girl 1.29. Here, again, we have a difference in favor of 
the girls, this time of 11%. A further comparison of the number 
of errors made by boys and girls, while not specially significant, 
shows a few points of interest. 8+4, with 25 errors, seemed to 
be much more difficult for the boys than for the girls, who made 
only 12 errors, while 9 + 6, with only 31 errors, seemed to be 
much easier for the boys than for the girls ? who made 51 errors. 
9 + 8, on the other hand, with 56 errors, seemed to be more diffi¬ 
cult for the boys than for the girls, who made only 39 errors in 
this combination. In all the cases, however, the percentages of 
error differ but slightly. 

















































61 


TABLE VIII. 


SHOWING ORDER OF DIFFICULTY OF ADDITION FACTS AND NUMBER OF ERRORS 
MADE BY 1,056 CHILDREN IN EACH FACT. 


Combination. 

No. Errors. 

Combination. 

No. Errors. 

9 + 8 (6) . 


7 + 7 

20 

9 + 7 (1) . 


6 + 6 . 


9 + 6 (3) . 


5 + 4 . 

. 20 

8 + 7 (8) . 


6 + 3 . 


8 + 5 (2) . 


7 + 1 

20 

8 + 6 (omitted) 

.... 66 

6 + 2.. 

. 19 

7 + 5 (10) . 

56 

8 + 1 . 


9 + 4 (omitted) 

.... 51 

3 + 1 . 


7 + 6 (15) . 

,... 50 

4 + 3 . 

. 18 

9 + 5 (5) . 


3 + 2 . 

17 

7 + 4 (13) . 

.... 48 

6 + 1 . 


9 + 3 (4) . 


1 + 1 . 


8 + 3 (14) . 

.... 41 

4 + 2. 


8 + 8 (12) . 

.... 37 

9 + 2. 

. 15 

8 + 4 (7) .. 

.... 37 

5 + 1 . 

. 15 

7 + 3 (11) . 

37 

4 + 1 . 


6 + 4 (omitted) 

34 

5 + 2. 


6 + 5 (9) . 

.... 32 

9 + 1 . 

. 13 

9 + 9 . 

,... 29 

8 + 2 ... 


5 + 3 .. 

,... 26 

5 + 5. 


7 + 2 . 

.... 24 

2 + 2. 

9 

2 + 1 .. 

,... 21 

4 + 4. 




3 + 3 . 



Note.—T he numbers in parenthesis indicate the order of the fifteen most difficult 
combinations according to the study made by Phelps (22) with eighth grade children. 


1. Discussion ,—In the above table it is interesting to note that 
the first 16 combinations in which errors are most frequent have 
each in them either a 9 or an 8 or a. 7, the 8 ? s appearing in the 
greatest number of cases, the 9’s next, but in only 3 fewer cases, 
and the 7 ? s next (cf. III. 1). It is also significant that the last 
four combinations on the list are doubles. That 1 + 1 stands so 
high in the number of errors appearing therein is rather startling, 
especially in the light of the fact that “1 + 1 = 1” was regarded 
as an error in “process” and not counted in this table. It is also 
interesting to observe the combinations which the same number of 
errors brings together; for example, 8+8, 8+4, 7+3; 7+7, 
6 + 6, 5 + 4, 6 + 3, 7 + 1, &c. 

In order to show the relation of the number of errors in “pro¬ 
cess” to the number of other errors, records were kept for 720 
third grade children as shown by the following table: 



















































TABLE IX. 


SHOWING THE NUMBER OF ERRORS MADE IN “PROCESS” (COLUMN “P”) AS COM¬ 
PARED WITH THE NUMBER OF OTHER ERRORS (COLUMN “o”) MADE BY 720 
THIRD GRADE CHILDREN. 


Combination. 

P 

O 

Combination. 

P 

O 

1. 

28 

12 

4. 

57 

3 

o 

39 

IS 

5. 

57 

13 

3. 

41 

14 

6 + 4. 

54 

18 

4. 

42 

38 

7. 

50 

33 

5 + 1. 

38 

12 

8. 

40 

17 

6.!. 

40 

12 

19. 

39 

31 


40 

13 




8. 

3S 

11 

5. 

41 

5 

i). 

40 

5 

1 6. 

56 

16 



| 7 + 5. 

46 

21 

9 

28 

7 

8. 

53 

52 

3. 

47 

n 

9. 

33 

28 

4. 

51 

10 




5. 

72 

9 

6. 

45 

15 

6 + 2. 

68 

11 

7 + 6. 

36 

29 

7. 

56 

1.3 

8.'. . 

45 

3f 

8. 

64 

8 | 

9 . 

32 

38 

9. 

52 

8 







7. 

35 

13 

3. 

59 

4 

8 + 7. 

35 

37 

4. 

70 

13 

9.. . 

33 

56 

5. 

54 

12 




6.. 

63 

8 

S + S. 

26 

23 

7 + 3. 

52 

18 

9. 

27 

53 

8. 

48 

17 



9.| 

52 

28 

9 + 9. 

31 

14 








Discussion .—The large number of errors in process is doubtless 
due to the fact that the emphasis of the year’s teaching had been 
upon the multiplication tables and not upon the addition facts. 
Tt might be argued that the total number of errors made in a given 
combination, counting both errors in process and other errors, gives 
a truer index of relative difficulty than the a other errors” as above 
used. In answer to this it will be observed that errors in process 
are generally most frequent when the multiplication of the num¬ 
bers involved is easiest, and that these errors decrease as the diffi¬ 
culty in multiplication increases. The total of both errors in pro¬ 
cess and “other errors” would make 4 + 2 and 4 + 3 the most 
difficult of all the facts, which is manifestly not the case. 











































































^ *1 ** ^ 










































































































































































































































































































































































































































































































































































64 


It may be wondered how an error in process was counted for 
2 + 2, in which the sum and the product are the same. This 
combination appeared in the group 7 + 9, 2 + 2, 3 + 1. . If the 
first and last answers of the group were obtained by multiplication, 
it was presumed that the middle one was also multiplied instead of 

added. # . 

Discussion .—The summary of results thus far obtained in addi¬ 
tion is shown in the graphic representation (Graph \). Of 
these “curves,” two should be especially helpful to the primary 
teacher: (1) the one giving the average co-efficients of difficulty 
for learning the combinations for both boys and girls; and (2) 
that showing the percentage of errors made by children. The 
former should indicate to the teacher the easy and the hard places 
as she proceeds in teaching the forty-five facts, while the latter 
shows the main points of attack in conducting review exercises 
with children who have presumably once learned the facts. 


V 


65 


TABLE X. 

SHOWING THE NUMBER OF ERRORS MADE IN THE MULTIPLICATION COMBINATIONS 

BY 1,215 CHILDREN. 


1 

J 

Boys. 

Girls. 

ns 4 » 

Per Cent. 

of Errors. 

Grade .... 

4 

3 

2 & 3 

5 & 3 

Totals. 

4 

3 

2 & 3 

5 & 3 

Totals. 

2 

C3 4-J 

^ O 

O H 

No. Pupils, 

133 

334 

61 

58 

586 

135 

394 

42 

58 

629 

1215 

.... 

*4. 

6 

13 

6 

' 18 

43 

6 

19 

4 

4 

33 

76 

6.2 

6 . 

4 

15 

6 

29 

54 

12 

23 

3 

10 

481 

102 

8.4 

8X3. 

10 

31 

7 

17 

65 

19 

40 

4 

17 

86 

151 

12.4 

9. 

13 

20 

10 

23 

72 

22 

47 

8 

20 

971 

109 

13.9 

12 . 

21 

40 

6 

16 

89 

19 

61 

2 

12 

94 

183 

15.0 

4. 

9 

16 

4 

7 

30 

10 

21 

2 

9 

42 

78 

7.2 

8 . 

22 

54 

6 

37 

119 

23 

59 

15 

19 

116 

235 

19.3 

9x4. 

39 

63 

r? 

i 

38 

147 

31 

76 

16 

22 

145 

292 

24.0 

10. 

9, 

1 


5 

8 

4 

19 



23 

31 

2 5 

12 . 

25 

60 

li 

26 

122 

20 

82 

8 

18 

128 

250 

20.6 

5. 


7 

i 

4 

12 

4 

14 


4 

22 

34 

2 8 

6 . 

14 

27 

6 

16 

63 

16 

39 

2 

18 

75 

138 

11.9 

7. 

14 

47 

10 

20 

91 

17 

50 

4 

19 

90 

181 

14.9 

8 . 

16 

28 

12 

19 

75 

13 

31 

3 

15 

62 

137 

11.3 

9x5. 

36 

16 

7 

28 

87 

25 

32 

5 

19 

81 

168 

13.8 

10. 

9 

12 


5 

26 

3 

28 


1 

32 

58 

4.8 

11. 

8 

99 

1 

13 

54 

13 

34 


12 

59 

113 

9.3 

12. 

39 

59 

5 

28 

131 

31 

90 

2 

17 

140 

271 

22.3 

6 . 

23 

26 

4 

15 

68 

20 

26 

4 

11 

61 

129 

10.6 

7. 

34 

64 

11 

28 

137 

30 

85 

8 

25 

148 

285 

23.4 

8 . 

42 

77 

13 

30 

162 

45 

100 

11 

24 

180 

342 

28.2 

9X6. 

47 

90 

15 

39 

191 

45 

113 

10 

31 

199 

390 

32.1 

10. 

11 

20 

1 

7 

39 

12 

22 

1 

5 

40 

79 

6.5 

11 

29 

38 

5 

15 

87 

16 

30 


11 

57 

144 

11.8 

12. 

35 

90 

11 

35 

171 

36 

114 

8 

32 

190 

361 

28,9 

7. 

31 

61 

3 

31 

126 

41 

81 

2 

18 

142 

268 

22.1 

8. 

54 

95 

11 

39 

199 

65 

134 

8 

29 

236 

435 

35.8 

9X7. 

53 

103 

12 

46 

214 

65 

131 

15 

30 

241 

455 

37.4 

10. 

13 

19 

1 

9 

42 

14 

22 

1 

r» 

i 

44 

86 

7.1 

11. 

22 

35 

1 

11 

69 

17 

39 

1 

11 

68 

137 

11.3 

12. 

54 

103 

14 

32 

203 

45 

147 

8 

35 

235 

438 

36.0 

8. 

51 

93 

5 

26 

175 

49 

109 

9 

19 

186 

361 

29.7 

9. 

54 

95 

18 

38 

205 

57 

US 

13 

29 

217 

422 

34.7 

10x8. 

13 

14 

2 

8 

37 

11 

34 

1 

2 

48 

S5 

7.0 


27 

49 

3 

14 

93 

19 

46 


9 

74 

167 

13.7 

12. 


116 

11 

33 

219 

52 

155 

5 

29 

241 

460 

37.9 

















































































66 


TABLE X.— Continued. 


SHOWING THE NUMBER OF ERRORS MADE IN THE MULTIPLICATION COMBINATIONS BY 

1.215 CHILDREN. 





Boys. 




Girls. 


r-J Xfl 

-M «j 

r-4 ?-* 

5 g 

O 2 

Grade .... 

4 

3 

2 & 3 

5 & 3 

Totals. 

4 

3 

2 & 3 

5 & 3 

Totals. 

m c3 

U O 

O Eh 

c 

No. Pupils, 

133 

334 

61 

58 

586 

1 

135 

394 

42 

58 

629 

1215 

.... 

9. 

34 

62 

4 

22 

122 

43 

74 

6 

18 

141 

263 

21.6 

10X9. 

11 

19 

2 

7 

39 

11 

40 

1 

3 

55 

94 

7.7 

11 . 

22 

45 

3 

12 

82 

25 

61 

2 

11 

99 

181 

14.9 

12 . 

57 

101 

5 

29 

192 

50 

143 

4 

28 

225 

417 

34.4 

10 . 

25 

61 

2 

6 

94 

32 

107 


8 

147 

241 

19.8 

11X10. . . . 

79 

170 

11 

32 

292 

69 

240 

5 

32 

346 

638 

52.5 

12 . 

61 

136 

5 

28 

230 

67 

209 

6 

30 

312| 

542 

44.6 

11 X11 .... 

80 232 

4 

35 

351 

85 

268 

2 

29 

as4 

735 

60.5 

12 . 

7S|1S1 

6 

40 

305 

83 

234 

6 

27 

350| 

655 

54.0 

12 x12. .. . 

48] 119 

1 

23 

191 

56 

163 


15 

2341 

425 

35.0 


* See note at bottom of Table VI.—Part I. 


It should be noted in connection with the above figures that the 
results from the fourth grade pupils were secured during the first 
week of school in September, so likewise those from the “5 & 3” 
grades. All other results were obtained in June. 














































67 


TABLE XI. 

9 

SHOWING ORDER OF DIFFICULTY OF THE MULTIPLICATION COMBINATIONS AND THE 
NUMBER OF ERRORS MADE BY 1,215 CHILDREN IN EACH FACT. 


Combination. 

No. Errors. 

Combination. 

No. Errors. 

11 X 11 . 


6X3. 

. 102 

12 X 11 . 


11 X 3 . 

. 99 

11 X 10 . 

.... 638 

10 X 9 . 

. 94 

12 X 10 . 

.... 542 

10 X 7 . 

86 

12 X 8 . 

.... 460 

10 x 8. 

85 

9 X 7 (1) ... 

.... 455 

12 X 2 . 

. 81 

12 X 7 . 

.... 438 

10 x 6. 

79 

8X7 (4) ... 

.... 435 

4x4 . 

. 78 

12 X 12 . 

.... 425 

4x3 . 

. 76 

9 X 8 (7) ... 

.... 422 

7X3 . 

. 71 

12 X 9 . 

.... 417 

10 X 5 . 

. 5S 

9 X 6 (2) ... 

.... 390 

8X2 ...... 

. 5S 

8X8 (3) ... 

.... 361 

5X4 . 

. 55 

12 X 6 . 

.... 361 

6x2 . 

. 50 

8x6 (5) ... 

.... 342 

5X3 . 

. 46 

9X4 . 

.... 292 

11 X 2 . 

. 46 

7x6 (6) ... 

.... 285 

1 X 1 . 

. 41 

12 X 5 . 

.... 271 

9x2 . 

. 39 

7 X 7 (S) ... 

.... 268 

10 X 3 . 

. 38 

9x9 (10) .. 

.... 263 

7x2 . 

. 38 

12 X 4 . 

250 

5x5 . 

. 34 

10 X 10 . 

241 

4x2 . 

. 32 

8x4 (9) ... 

.... 235 

10 X 4 . 

. 31 

7X4 . 

.... 192 

10 X 2 . 

. 31 

12 x 3 . 

183 

11 x 1 . 

. 31 

11 X 9 

181 

4X1 . 

. 31 

7x5 

181 

3X1 . 

. • 28 

9x3 

169 

5x2 . 

. 26 

9 y 5 

168 

3x3 . 

. 25 

ii v 8 

167 

9X1 . 

. 22 

8 v 3 

151 

3X2 . 

. 21 

11 V fi 

144 

7X1 . 

. 21 

-L-L A v . 

v 5 

138 

6X1 . 

. 21 

11 v 7 

137 

12 X 1 . 

. 20 

ft v 5 

137 

5X1 . 

. 20 

ft y 4 

133 

2X1 . 

. 20 

11 v 4 

131 

2X2 . 

. 18 

a v 

129 

8X1 . 

. 18 

11 X 5 . 

113 

10 X 1 . 

. 12 


N OTE> _The numbers in parenthesis indicate the order of the ten most difficult 

combinations according to the study made by Max Doling (6). 


Discussion .—Table X shows the errors which were made in 
the same kind of a test in multiplication as was given in addi¬ 
tion. It represents the tabulation of over 13,000 errors. The 
average number of errors for each boy was 10.8, that for each girl 

























































































68 


11.2. As this was essentially a test in power of retention, the 
boys seemed to possess that power in the case of multiplication 
2.7 % in excess of the girls. A comparison of the details shows 
little or-no difference between the boys and the girls until 11 X 8 
is reached. From there to 12 X 12 the percentage of errors 
made by the girls is constantly higher than that made by the boys. 
A comparison of the tables in which 7, 8, 9, and 12 appear as the 
first factor shows practically no difference until 12 is reached. 
Here the difference is marked, being nearly 10% in the favor of the 
boys. With the 10th and 11th rows the advantage is also with the 
boys. With the 6th and below the advantage is slightly with the 
girls, but it is so small as to be almost disregarded. It would 
seem, therefore, that boys have a retentive power for larger prod¬ 
ucts in excess of that possessed by the girls. 

As has been already explained, Professor Doring’s experiment 
extended only to 10 X 10. A comparison in Table XI with the 
order established by his investigations shows that only 9X4 and 
10 X 10 of the numbers coming under his consideration crept into 
his ten most difficult combinations, and that only the first place 
of difficulty in' the two experiments agree, though a general 
agreement as to relative order of the other places, with the ex¬ 
ception of his 4th and 7th places of difficulty, is to be noted. 
One woufd think that 10 X 10 would have very few errors, but 
the combination appeared so often as “10 X 10 = 110.” The 
“10” sequence established in the part of the problem carried 
over with disastrous results into the product. 


2. ERRORS m PROCESS. 

Errors in process were not counted in the multiplication tests 
just described, for the same reasons given for not counting them 
in the addition tests. However, in order to show the relation 
which the number of such errors bore to other errors the fol¬ 
lowing table constituting records for 728 third grade pupils is 
added: 


69 


TABLE XII. 

SHOWING TIIE NUMBER OF ERRORS MADE IN “PROCESS” (COLUMN “p”) AS COM¬ 
PARED WITH THE NUMBER OF OTHER ERRORS (COLUMN “o”) MADE BY 728 
THIRD GRADE CHILDREN. 


Combination. 

P 

O 

Combination. 

P 

O 

1. 

94 

31 

5. 

40 

21 

2. 

43 

16 

6. 

25 

66 

3. 

33 

23 

7. 

22 

97 

4. 

22 

26 

8. 

20 

59 

5 . 

29 

12 

9. 

20 

48 

6 x 1 . 

27 

16 

10. 

19 

40 

7. 

26 

15 

11.' 

19 

56 

8. 

25 

8 

12 X 5. 

20 

149 

9. 

22 

15 




10 . 

22 

8 

6. 

38 

52 

ii . 

20 

21 

7. 

22 

149 

12 . 

24 

15 

8. 

23 

177 




9. 

24 

203 

2 . 

25 

8 

10. 

21 

42 

3 . 

34 

6 

11. 

22 

68 

4 . 

35 

14 

12 X 6. 

24 

204 

5. 

24 

13 




f\ . 

29 

14 

7. 

42 

142 

7 v 2 

21 

13 

8. 

24 

229 

8 . 

23 

19 

9x7. 

22 

234 

Q .... 

23 

7 

10. 

21 

41 

10 . 

21 

17 

11. 

20 

74 

11 

21 

25 

12. 

20 

250 

12. 

24 

44 


1 





■ 8X8. 

37 

202 

o 

68 

12 

9. 

22 

213 

A 

31 

32 

10. 

22 

48 

K 

27 

20 

11..... 

20 

95 

C 

29 

38 

12. 

21 

271 

7. 

26 

28 




Q w Q 

24 

77 

9. 

27 

136 

o X o.• 

Q 

24 

73 

10. 

19 

59 

1 o 

20 

18 

11. 

17 

106 

11 ... 

17 

53 

12 X 9.. 

20 

244 

12 . 

22 

107 







10 X 10 . 

40 

168 

4 

61 

37 

11 X 10 . 

21 

410 


23 

28 

12 X 10. 

22 

345 

... 

6. 

19 

62 




7 

20 

95 

11 X 11 . 

53 

500 


24 

113 

12 X 11 . 

19 

415 

9. 

18 

139 




10 

20 

20 

12 X 12. 

42 

2S2 

11. 

18 

82 




12 X 4 . 

23 

| 142 

1 














































































































70 


Discussion .—An examination of the above table shows that the 
predominance of errors is found in the doubles, especially in 
1 X 1. The small number in 2X2, compared with the other 
doubles, is due, of course, to the fact that the product and the 
sum of this combination are the same, so it was frequently im¬ 
possible to tell from the context whether the pupil had added 
or multiplied. 

Here again it was thought best to count only errors which 
could not be subsumed under the head of process or of inversions. 
The number of inversions was so small, however, as to be neg¬ 
lected, there being only 22 in all, 8 of these being in 9 X 5 and 
4 in 9 X 6? which, of course, is significant. 

Discussion ,—The summary of the results so far obtained in mul¬ 
tiplication is shown in the above graphic representation (Graph 
VI). The most interesting thing noticed in a study of the three 
“curves” recorded in this figure is their similarity and their dispar¬ 
ity. Up to 10 X 5 the similarity between the values of the total er¬ 
rors co-efficients of difficulty and the per cent, of errors made by 
1,215 children is very striking indeed. At this point a diversion 
begins which constantly increases to the end of the scale, yet coin¬ 
ciding in direction in almost every point. Up to 10 X 3 the per 
cent, values are almost invariably lower than the total errors co¬ 
efficients of difficulty values. From 10 X 3 to 10 X 5 there 
seems to be a struggle for supremacy, as it were, in which first 
one of them then the other succeeds, but after 10 X 5 the per 
cent, values are almost invariably higher. This, of course, results 
from the relation which exists between the two kinds of co-effi¬ 
cients shown in the figure, the learning co-efficient of difficulty 
having in it the same elements, though differently treated, as 
are contained in the total errors co-efficient. 

The most significant relationship, however, lies in the com¬ 
parison of the learning co-efficients and the per cent, of error. 
Up to 10 X 4 the former a^e invariably higher than the latter; 
then comes again a struggle for supremacy, as it were, lasting to 
about 10 X d. After that point, with one notable exception, that 
of 9 X 7, the learning co-efficients are lower to the end of the 
series. While their divergence constantly increases, the value 


















































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































/ 




71 


tendencies in both cases remain, .for the most part, in the same 
direction. 

The high value of the learning co-efficient in the early part 
of the scale, even for what we usually regard as the easy com¬ 
binations, is commensurate with the time taken for mastery and 
the number of errors made in the process of mastery. The low 
values at the end of the scale are likewise commensurate with 
the same factors, and demonstrate beyond question the develop¬ 
ment of power in learning the combinations in the early part of 
the number scale. 

The relation between the two “curves” would seem to show 
that facts easily mastered are easily forgotten. Compare the 
“curves” at the points 8X4, 9X4, 9X6, 9 X T, 12 X 8, 
12 X 9, 11 X 10, 11 X 11, and 12 X 11. 


As in the case of Graph V for addition the learning co-effi¬ 
cients of difficulty here taken in the order and in the groups in 
which the combinations were presented should be especially help¬ 
ful to the teacher in teaching the facts, and likewise the per 
cent, of errors “curve” should be of great value to the teacher 
in directing the review work of her class. 


XV. THE EFFECT OF SUMMER VACATION UPON MEM¬ 
ORY FOR THE COMBINATIONS. 

In order to determine the effect of ten weeks of summer vaca¬ 
tion upon the memory for number combinations, the same tests 
were given during the first week of school in September as were 
given during the last week of school in June, but to classes one 
in advance of those in which the test in June was given. This 
gave results only for such pupils as were promoted. Five hundred 
and fifty-eight pupils were involved in the tests, distributed as 
follows: 516 third grade, 20 second grade, and 22 first grade. 
The entire results are given below: 




72 


TABLE XIII. 

SHOWING THE NUMBER OF ERRORS IN ADDITION MADE BY 558 PUPILS BEFORE AND 

AFTER SUMMER VACATION. 



Before Vacation. 

After Vacation. 


Grade . 

3 

2 

1 

Total. 

3 | 

2 

1 

Total. 

Increase. 

1 . 

2 

1 


3 

2 

4 


6 

3 

2 . 

5 

2 

1 

8 

3 

6 


9 

1 

3 . 

4 

3 


7 

9 

6 


15 

8 

4 . 

4 

1 


5 

2 

5 


7 

2 

5 + 1 . 

8 

1 


9 

1 

5 


6 

— 3 

6 . 

5 



5 

7 

5 


12 

7 

7 . 

5 

1 


6 

5 

3 

1 

9 

3 

8 . 

•7 

1 


8 

2 

5 


7 

— 1 

9 . 

3 

2 


5 

5 

3 

1 

9 

4 

2 . 

2 



2 

2 

9 


4 

2 

3 . 

6 

1 


7 

7 

4 

1 

12 

5 

4 . 

5 

2 


7 

9 

2 

2 

6 

— 1 

5 . 

6 


1 

7 

6 

6 

2 

14 

7 

G + 2 . 

5 

1 

1 

7 

6 

6 

2 

14 

7 

7 . 

6 

2 


8 

7 

4 

1 

12 

4 

8 . 

5 

1 


6 

7 

4 

1 

12 

6 

9 . 

5 

2 

. 

r* 

< 

6 

7 

4 

i x “ 

17 

10 

3 . 

1 

0 


1 

3 

4 


7 

0 

4 . 

6 

1 

1 

8 

4 

5 

3 

12 

4 

5 . 

12 

1 

1 

14 

10 

r* 

4 

1 

18 

4 

6+3 . 

3 

1 

1 

5 

12 

5 

4 

21 

16 

7 . 

16 

2 

1 

19 

18 

5 

3 

26 

7 

8 . 

11 

3 

1 

15 

18 

6 

1 

25 

10 

9 . 

14 

2 

1 

17 

19 

8 

4 

31 

14 

4 . 




0 

4 

1 


5 

5 

5 . 

6 

2 


8 

9 

6 

5 

20 

12 

6 + 4 . 

10 

1 


11 

14 

6 

1 

21 

10 

7 . 

18 

2 

2 

22 

24 

5 

3 

32 

10 

8 . 

9 

2 

3 

14 

13 

5 

1 

19 

5 

9 . 

23 

1 

3 

27 

25 

9 

7 

41 

14 

5 .. 

2 

1 


3 

4 

1 


5 

9 

6 . 

12 

1 

1 

14 

17 

8 

4 

29 

15 

7 + 5 . 

14 

2 

1 

17 

34 

5 

5 

44 

27 

8 . 

39 

4 


43 

30 

5 

4 

39 

_4 

9 . 

IS 

2 


20 

29 

10 

2 

41 

21 

6 . 

5 

T » » » * 

1 

6 

5 

4 

1 

10 

4 

7 + 6 . 

22 

3 

3 

28 

23 

9 

5 

37 

9 

8 . 

21 

1 

4 

26 

36 

8 

3 

47 

21 

9 . 

29 

1 

1 

31 

38 

10 

6 

54 

23 

















































































































73 


TABLE XIII.— Continued. 

SHOWING THE NUMBER OF ERRORS IN ADDITION MADE BY 558 PUPILS BEFORE AND 

AFTER SUMMER VACATION. 



Before 

Vacation. 


[fter Vacation. 


Grade . 

3 

2 

1 

Total. 

3 

2 

1 

Total. 

Increase. 

7 . 

3 


1 

4 

r* 

i 

7 

2 

16 

12 

8+7 . 

29 

2 

3 

34 

32 

9 

5 

46 

12 

9 . 

30 

4 

2 

36 

. 45 

9 

4 

5S 

22 

8+8 . 

10 

1 

1 

12 

12 

7 

3 

22 

10 

9 . 

38 

4 

3 

45 

36 

13 

6 

55 

10 

9 + 9 . 

9 

1 


10 

13 

6 

1 

20 

10 


TABLE XIV. 

SHOWING ADDITION FACTS ARRANGED IN THE ORDER OF THE PER CENT. FORGOTTEN 

DURING THE SUMMER VACATION. 


Combination. 

Loss Per Cent. 

Combination. 

Loss Per Cent. 

7 4-5 

4.85 

5 + 2 . 

. 1.3 

• “p U • • • • • 

9 + 6 . 

. 4.1 

6 + 1 . 

. 1.3 

9 + 7 . 

. 3.9 

3 + 3 . 

. 1.1 

8 4-6 

3.8 

8 + 2 . 

. 1.1 

9 + 5 . 

. 3.8 

8 + 4 . 

. 0.9 

6 + 3 . 

. 2.9 

4 + 4 . 

. 0.9 


2.7 

3 + 2 . 

. 0.9 

9 + 4 . 

. 2.5 

6 + 6 . 

. 0.7 

9 + 3 . 

. 2.5 

5 + 3 . 

. 0.7 

c _i_ 7 

2.15 

4 + 3 . 

0.7 

o -p 1 • • • • • 

7 4- T 

2.15 

7 + 2 . 

. 0.7 

• 1 • • • • • * 

5 + 4 . 

. 2.15 

7 + 1 . 

. 0.7 

9 + 9 . 

. 1.8 

7 + 1 . 


9 + 8 . 

. 1.8 

1 + 1 . 

. 0.5 

O 1 c 

1.8 

5 + 5 . 

. 0.4 

7 + 4 . 

. 1.8 

2 + 2 . 

. 0.4 

6 4-4 

1.8 

4 + 1 . 

. 0.4 

8 + 3 . 

. 1.8 

2 + 1 . 

. 0.2 

Q 1 O 

1.8 

4 + 2 . 

. — 0.2 

7 + 6 . 

. 1.6 

8 + 1 . 

.— 0.2 

o 1 1 

1.4 

5 + 1 . 


rr i o 

1.3 

8 + 5 . 

.—0.7 

( + O . 

6 + 2 . 

. 1.3 




Note _ The n eo-ative values indicate gain instead of loss. Percentages are cal¬ 

culated 'on the basis of the number of errors possible. 





















































































74 











































































































































































































































































































































































































































































































































































































75 


Discussion.—In the study of Graph YII and Table XIV a 
word of warning is necessary. They alone should not be made 
the basis of any teaching practice. One point in the figure and 
table will serve to illustrate. It will be noted that at 8 + 5 there 
is an actual gain in knowledge, indicated by the “curve’s” fall¬ 
ing below the zero line, while the per cent, of loss for 7 -f- 5 is 
the maximum for the whole figure. This does not mean that in 
the process of review of the facts of addition that much em¬ 
phasis should be placed upon the 7 + 5 fact and little or no 
attention should be given to the 8 + 5. The same point is to 
be noted with the high and the low values throughout the “curve.” 
In the case above cited only 17 errors appeared for 7 + 5 in 
the June tests while there were 43 for 8 + 5. Greater oppor¬ 
tunity for loss was, therefore, presented by the former than for 
the latter. In the September tests there actually occurred five 
fewer errors for 8 + 5 than for 7 + 5. This figure and table 
should be studied in connection with Table VII and the cor¬ 
responding part of Graph V. 

The value of Table IX and Graph VII lies in this fact: 
they point out the relative stability of the combinations from 
the standpoint of memory. 


76 


TABLE XV. 

SHOWING THE NUMBER OF ERRORS MADE BY 530 THIRD GRADE CHILDREN IN THE 
MULTIPLICATION COMBINATIONS BEFORE AND AFTER THE SUMMER VACATION. 


Combinations. 

Before. 

After. 

Increase. 

Combinations. 

Before. 

After. 

Increase. 

Combinations. 

1 . 

22 

28 

6 

8X3.... 

55 

85 

30 

11X6. ... 

2 . 

11 

12 

1 

9. 

58 

81 

23 

12 . 

3. 

12 

5 

— 7 

10 . 

17 

26 

9 


4. 

19 

3 

—16 

11 . 

33 

52 

19 

7. 

5. 

7 

6 

— 1 

12 . 

70 

93 

23 

8 .. 

6X1.... 

12 

8 

—4 



9X7... . 

7 . 

9 

5 

— 5 

4 . 

30 

27 

— 3 

10 . 

8 . 

3 

5 

2 

5 . 

19 

30 

11 

11 . 

9 . 

10 

9 

—1 

6 . 

47 

69 

22 

12 . 

10 . 

4 

6 

2 

7 . 

71 

105 

34 


11 . 

16 

17 

1 

8x4.... 

67 

121 

54 

8 . 

12 . 

s 

8 

0 

9 . 

102 

161 

59 

9x8. 





10 . 

i 

21 

14 

10 . 

2 . 

6 

9 

3 

11 . 

51 

41 

— 13 

11 

3. 

6 

17 

11 

12 . 

103 

130 

27 

12 .. 

4. 

12 

7 

— 5 





5 . 

6 

12 

6 

5 . 

9 

13 

4 

9x9 

6 . 

7 

25 

18 

6 . 

45 

69 

24 

10 . 

7X2. ... 

6 

18 

12 

7 . 

74 

80 

6 

11 . 

8 . 

13 

35 

22 

8x5. .. . 

39 

76 

37 

12 . 

9 . 

4 

20 

16 

9 . 

44 

1041 HO 


10 . 

13 

13 

0 

10 . 

26 

29 

— 4 

10 

11 . 

15 

17 

2 

11 . 

29 

391 10 

11 x 10 

12 . 

24 

40 

16 

12 . 

96 

144| 48 

12 . . 

3. 

6 

11 

5 

6 . 

34 

1 

71] 37 

11 x 11 

4v3. 

22 

30 

8 

r* 

( . 

108 

151| 43 

12 . 

5. 

14 

21 

7 

8 . 

124 

170| 46 


6 . 

30 

49 

19 

9x6.... 

142 

196| 54 

12 x 12 ... 

7. 

18 

46 

28 

10 . 

23 

40| 17 



Note.— Average time taken for test before vacation, G.3 minutes; 
5.9 minutes. 


42| 76 

34 

130| 180 

50 

9lj 141 

50 

15lj 249 

98 

166 256 

90 

33 

35 

2 

46 

69 

23 

164 

226 

62 

143 

203 

60 

151 

253 

102 

31 

28 

—3 

59 

81 

22 

165 

260 

95 

91 

184 

93 

42 

42 

0 

70 

90 

20 

146 

270 

124 

105 

117 

12 

263 

318 

55 

227 

290 

63 

326 

356 

30 

278 

341 

63 

188 

241 

53 


after vacation, 

































































































77—a (?) 






























































































































































































































































































































































































































































































































































































































































77 


TABLE XVI. 


SHOWING MULTIPLICATION COMBINATIONS IN THE ORDER OF THE PER CENT. 
FORGOTTEN DURING THE SUMMER VACATION. 


Combination. 

Loss. 

Combination. Loss. 

Combination. 

Loss. 

12 

X 

9 . 

% 

. 23.4 

8 

X 

3 ... 

% 

7 

x 

5 . 

% 

1 i 

9 

X 

8 . 

. 19.2 

7 

X 

3 ... 

.5.3 

5 

x 

9 

1 i 

8 

X 

7 . 

. 18.5 

12 

X 

4 ... 


1 

X 

1 . 

1.1 

12 

X 

8 . 

. 17.9 

6 

X 

5 ... 

.4.5 

3 

X 

3 . 

. 0.9 

9 

X 

9 . 

. 17.5 

11 

X 

7 ... 

.4.3 

5 

x 

5 

0 75 

9 

X 

7 . 

. 17.0 

12 

X 

3 ... 

.4.3 

2 

X 

2 

. 0.6 

12 

X 

11. 

. 11.9 

! 9 

X 

3 ... 

.4.3 

10 

X 

7 . 

0.4 

12 

X 

10. 

. 11.9 

n 

X 

8 ... 

.4.15 

11 

X 

2 . 

0.4 

12 

X 

7 . 

. 11.7 

6 

X 

4 ... 

.4.15 

10 

X 

1 . 

. 0.4 

8 

X 

8 . 

. 11.3 

8 

X 

2 

. 4.15 

8 

x 

1 

0 4 

9 

X 

5 . 

. 11.3 

11 

X 

9 ... 


11 

X 

1 . 

. 0.2 

9 

11 

X 

4 . 

. 11.1 

11 

X 

3 ... 

.3.6 

2 

X 

1 . 

. 0.2 

X 

10. 

. 10.4 

6 

X 

3 ... 

.3.6 

10 

X 

9 . 

0 

9 

X 

6 . 

. 10.2 

6 

x 

2 

.3.4 

10 

x 

2 

o 

8 

X 

4 . 

. 10.2 

10 

X 

6 ... 

.3.2 

12 

X 

1 . 

0 

12 

X 

12. 

. 10.0 

12 

X 

2 

.3.0 

4 

X 

1 . 

. —0.2 

7 

X 

7 . 

. 9.45 

9 

x 

2 

.3.0 

5 

V 

1 

_n 2 

12 

X 

6 . 

. 9.45 

12 

X 

4 ... 


10 

A 

X 

8 . 

• v . w 

. —0.6 

12 

X 

5 . 

. 8.85 

10 

X 

10... 

.2.3 

4 

X 

4 . 

. —0.6 

8 

X 

6 . 

. 8.7 

7 

X 

2 

.2.3 

10 

X 

5 . 

. —0.75 

7 

X 

6 . 

. 8.3 

5 

X 

4 ... 

.2.1 

7 

X 

1 . 

. —0.75 

6 

X 

6 . 

. 7.0 

3 

X 

2 

.2.1 

6 

X 

1 . 

. —0.75 

8 

X 

5 . 

. 7.0 

11 

X 

5 ... 

. 1.9 

4 

X 

2 . 

. —0.9 

11 

X 

6 . 

. 6.4 

10 

X 

3 ... 

. 1.7 

3 

X 

1 . 

. —1.3 

7 

X 

4 . 

. 6.4 

4 

X 

3 ... 


11 

X 

4 . 

. —2.45 

11 

X 

11 . 

. 5.65 

5 

X 

3 ... 

. 1.3 

4 

X 

1 . 

. —3.2 


Note. —The negative values indicate gain instead of loss. Percentages are cal¬ 
culated on the basis of number of errors possible. 


Discussion ,—It is interesting to compare the order in Table 
XVI with that arranged on the basis of number of errors made by 
1,215 pupils, shown in Table XI. It will be observed in this com¬ 
parison that the 21 combinations which stand highest in Table 
XVI are to be found among the first 23 combinations that appear 
first in Table XI. The next group of 10 in Table XVI is to be: 
found, with a few exceptions, in the corresponding part of Table 
XI, and so on, but there is no other similarity in the tables that is 
apparent. 

The ratio between the per cent, of loss to the per cent, of error is 
far from constant. If the value representing the lack of stability 
of the number facts bore a direct relation to the number of errors, 
this relation should be constant throughout. 


























































































78 


The same warning to which attention was called in regard to 
the application of Graph VII must also be issued here with refer¬ 
ence to the use of Graph VIII. It must be studied in connection 
with Graph VII, especially with reference to that part showing 
the per cent, of errors made by 1,215 children. 


XVI. VACATION SCHOOLS. 

1. RESULTS. 

Of the 558 children included in the addition tests some over 100 
attended school from one to four weeks during the summer. That 
this fact did not materially, if at all, affect the results for the whole 
group is shown by the following: The ratio between the number 
of errors made in June to the number made in September by 112 
pupils selected at random from those who did not attend vacation 
school was found to be .896 ; while a similar ratio for 112 children 
who attended vacation school was .815. What would have been the 
ratio for those who went to vacation school, if they had not at¬ 
tended is impossible to say, but whatever may have been the effect 
of instruction during the short summer period, it was not suffi¬ 
cient in this particular to bring the standing of those who took ad¬ 
vantage of it up to that made by those who did not. 

In the comparative multiplication tests the ratio of the number 
of errors in June to the number of errors in September for 112 
pupils, selected at random from those not attending vacation 
school, was found to be .735; while a similar ratio for the same 
number of children who did attend vacation school was .716, the 
results again being in favor of those children who did not go to 
school in summer. 

Since all the pupils in each case were promoted to the next 
higher class, we might presume that the vacation school was made 
up largely of those who may have stood low in their class. In any 
case, the fact that one-fifth of the pupils did attend school from one 
to four weeks during the summer cannot greatly influence the re¬ 
sults of these comparisons. 


79 


XVII. RELATIVE DIFFICULTY AS INDICATED BY SKILL 
IN MANIPULATION. 

1. PLAN AND NATURE OF THE TESTS. 

There remains but one other important test for the determina¬ 
tion of relative difficulty, and that is the test of skill in the use of 
the combinations. To determine this the forty-five addition com¬ 
binations were divided into nine groups of five facts each. These 
groups contained the same combinations as the groups used for 
teaching the facts in the first part of this investigation. The 
seventy-eight multiplications were divided into sixteen groups in 
like manner, except that the fourth group was inadvertently made 
to contain four combinations instead of the second group. 

For conducting this part of the experiment the combinations 
were arranged across a sheet of paper as shown below, the same 
being a reproduction of Group 3 for addition: 

( 3 ) 27242722 4 2326 

5232623526242 


There were usually about 15 examples in all, the purpose being 
to have more than any one pupil could do during the time allotted 
for each group. It will be observed that after the first five facts 
in the row the same facts are repeated in different order and in 
reverse form and direct form, so that no order or answers could 
assist the pupil in writing the products more rapidly than his 
normal rate would be for the first five sums or products, as the case 
might be. The nine groups were thus arranged for the first test in 
regular order down the page. For the second test the groups were 
arranged in reverse order, and for the third test the following 
order was given: 6, 7, 8, 9, 1, 2, 3, 4, 5. The object here was to 
equalize the effects of fatigue. Across the top of each sheet was 
typewritten, “Write the sums of the following numbers from left 
to right.” 

On the back of each sheet were mimeographed the answers of 
the various combinations in the order in which they appeared on 
the face, with a short line drawn under each. 

These tests were given to the A and B classes of the third and 
fourth grades. 




80 


2. THE OBJECT OF THE TESTS. 

The object of the experiment was first to determine the time re¬ 
quired to write the sums of the various groups, or the products, as 
the case required, on the face of the sheets, then to determine the 
time required to copy the answers on the hack of the sheets. By 
subtracting the time for the latter almost purely mechanical part 
from the former, a time representing more nearly.the net time re¬ 
quired to think the results would be obtained. That group requir¬ 
ing, therefore', the longest time to think out was adjudged the most 
difficult. 

3. MANNER OF CONDUCTING THE TESTS. 

The experiments were conducted personally by the writer and in 
the following manner: To each pupil there was given a sheet of 
white paper on which were mimeographed the combinations as 
above described. Covering each white sheet was a blank yellow 
sheet of the same size. This completely hid from view the work to 
be done. It was then explained that on the white sheet there was a 
number of rows of little examples in addition (or multiplication), 
the answers to which were to be written as fast as possible on the 
giving of a certain signal. The arrangement of the examples was 
shown on a sheet in the hands of the experimenter. He then showed 
how the yellow sheet was to be pulled down to disclose one row at a 
time at the signal, “papers down,” and how on the signal of- 
“write,” given immediately after, the pupils were to write as many 
of the answers as they could until the experimenter said “stop,” 
after which the papers were to be pushed up over the row just writ¬ 
ten. This was to prevent additions or corrections. At the next signal 
“papers down,” the next row below was to be exposed and as many 
answers in that row were to be written as time permitted. A rest 
of from five to ten seconds was given between the writing of each 
row. When the last row had been written, the papers were turned 
over and all the answers on the back of the sheet were exposed at 
once. It was then explained that these were to be copied in the 
place indicated by the mark under each and from left to right 
across the page, the work to begin at a certain signal. The signals 
used for this part of the test were as follows: (1) “Get ready,” 
which meant put the pencil where the first answer was to be 
written; (2) “write,” which meant that as many of the answers as 
possible were to be copied until the third signal, “stop,” was given. 
Five or ten seconds’ rest was allowed between each row. The order 


81 


of these groups of answers was made to follow the order of the 
groups of examples on the other side of the sheet. 

For the third grade addition tests 15 seconds was the time 
allowed for writing the sums of each group, and 5 seconds for the 
answers. For the fourth grade 10 seconds and 5 seconds, respec¬ 
tively, were allowed. 

4. METHOD OF EVALUATING RESULTS. 

The number of sums or products written, and the number of 
answers copied, were arranged in the tables of frequency which 
are given below. In these the horizontal row of figures at the top 
of the table represents the groups involved, the vertical row on the 
left-hand margin represents the number of results written or 
answers copied. At the bottom of each table is given the average 
number of results written and the average number of answers 
copied for the various groups. On this basis is calculated the 
number of seconds required to write the five results or to copy the 
five answers of the group. The difference between these values for 
each group gives the number of seconds required to “think” the 
sums or products. 

In order to test the validity of the results obtained, the average 
deviation is calculated for each of the averages; the standard de¬ 
viation, the P. E. and the P. E. in seconds required to do five 
combinations are calculated for the maximum and minimum aver¬ 
age deviations in the sums and products, and for the same groups 
in the answers copied. For the method by which these calcula¬ 
tions were made the writer is indebted to Professor E. L. Thorn- 
dyke (44). For calculating the P. E. in seconds to do five com¬ 
binations, the following proportion was used: The P. E. in seconds 
required to do five sums or products or to copy five answers (x) : 
the number of seconds required to write five sums, &c. :: the P. E. 
in combinations written in 5 seconds : the number of combiner 
tions that are written in 5 seconds. This will appear in the first 
calculation of the kind made as follows: 

x sec. : 10.86 sec. :: .28/3 sums, &c. : 7.24/3 sums, &c., a 
solution of which gives .4 as the P. EL in seconds to write five sums. 
The true average will, therefore, be seen not to vary materially 
from the calculated average in any of the cases as far as time is 
concerned. For a full discussion of the reliability of averages, the 
reader is referred to Dr. Thorndyke’s book, Mental and Social 

Measurements , already referred to above. 

6 


• '' i ■ TABLE XVI. 

SHOWING FOR A AND B CLASS PUPILS OF THE THIRD GRADE THE NUMBER OF SUMS IN THE VARIOUS GROUPS WRITTEN IN 15 SECONDS, THE 
NUMBER OF ANSWERS COPIED IN 5 SECONDS, AND GIVING THE TIME REQUIRED TO “THINK'’ THE ANSWERS TO THE SAME GROUPS. 


82 


X 

P5 

X 

£ 

<1 


H 

PA 

< 

& 


cs 

pq 

. . • b* o O CO 1£5 o • • .. 

• • • H Cl H rH. 

CD rb CO • • • 

LO Hb 7—! • * * 

< 

• • • • lO r1 Cl M O b- Cl • iH • • 
.... t-H Ol Ol tH • .... 

L— b- Ol • • • 

LO CO tH • • • 

CO 

pq 

• • t—! CO Ol CO GO GO CO 10). 

• • ••••••• 

t * • •••••• 

CO Ol 

GO Ol fN 
CO L— Ol LO tH tH 

IO Tjl t— 1 rl C O 

< 

• • • Ol • LO b- O H LO LO <M • <01 tH • • 
... *1—17—1 (01 rH • •• 

rH LO- LO • • • 

b- CO T-H • • • 

l- 

pq 

. I -H^COHOGO • • 01 • rH • • • 

••• HHNrl •• • ... 

c: oi oi • • • 

LO rb 7—1 • • • 

<1 

• • • •T-HLOCOTbCOrHOl • <01 00 rH • • 

.... t-H Ol t-H • •• 

cs oi go • • • 

b* CO T—1 • • • 

CO 

pq 

.. • H N Cl • ♦ • • 

Tb Ol Tb • • • 

CO CO t-H • • • 


’ * • • Ol LO 00 Ol Tb tH CO CO CO Ol LO <01 • 

•••• Ol t-H tH • 

t— 

CO OS' b- 
OCCCOHC 

GO CO Ol -Ol o o 

lo 

AH 

• ! ^ • CO GO C I'* LO Cl IO CO 01 • rH * 

.... Ol t-H •• 

1 

1 7.4 

: 3.4 

2.0 

< 

’ < ^ • • CO Ol O' Ol CD CD CO Tb (01 Ol b- ^ 

• • • • • tH tH rH tH • 

CO t— o • • • 

CS Ol Ol • • • 

^b 

pq 

• • 'HCOCCGOCIHH^tCOrH • • • • 

• • • rH rH rH tH • • • • 

Ol »C LO • • • 

b* CO rH • • • 

< 

• ^ • • -t-HOIQCiLDLDCOCSCOLD • CS • 

. n H rl • • 

Gw LG 1 O • • • 

Gw C1 • • • 

• • • 

CO 

pq 

• • • CO •COrHO^'^tCCrHiO • • • • 

• • • • tH rH rH rH • • • • 

O CC Kt • • • 

CO rH • • • 


. * . . rH Ol CS Ol LO O Ol CO rb Ol • b- 

. H rH H ri • 

rH lO <0 • • • 

CMC1 • • • 

rH • • • 

01 

pq 

• *01 • r"1 tH '’+< CS GO Ol ^b • H H • • 

.. » 7—! Ol rH • ... 

L— t— CO • • • 

CO CO tH • • . 


• • •HHCUO'KlOL-^OHClHb- • 

• • • t-H t—1 tH t—1 • 

tH o • • • 

CS Ol Ol • • • 

t-H 

pq 

• • • • CO lO 00 tH tH 01 lO CO 01 • • • • 

.... tHOItH .... 

Hb Tb LO • • • 

b* CO tH • • • 

< 

.HClCO^TtiOOlO^^C • 

. rH t— 1 rH t-H t-H • 

CO' rb Ol • • • 

o oi oi • • • 

tH ... 

Groups ... 

Classes . 

Number written— 

0 . 

1 . 

2 . 

3 . 

4 . 

5 .. 

6 . 

7 . 

8 . 

9 . 

10 . 

11 . 

12 . 

13 . 

14 . 

15 ... 

16 . 

Averages . 

No. seconds to copy 5 answers . 

Average deviation . 

Standard deviation .. 

P. E . 

P. E. in seconds to write 5 answers .. 


* 






































































































TABLE XVI —Part 2. Sums. 



a 


CC 




(NOiMIOOOOONHW 


McOHfflcxoo th oi 


fHO ^ ^ H -ti O 'M H 

M rH rt tH rH 


<1 

• rH b- O O O OI CO lO lO OI 01 01 • • rH • 

• OI H H H • • 

CO rH LO CO © • • • 

lO rp co © rH • • • 

rH rH ... 

PQ 

01©COCOHPrHrH , 0 , rH • rH rH • * • * • 

rH rH rH tH • • • • • • 

rH OI OI rp lO • • • 

hP 00 HP rH rH • • • 
rH ... 


t- 01 hP GO CO 


01 th 


^Ol-MC5 
HP l- CO CO rH 


Ci 

b- 01 b- no CC b- 


■*p © 

tH l— 


W C 't lO H H 
01 tH 


go 


PQ 


rHlOCOlOOlOnOCOnOOlrHrHHPrHOl 


GO t- Cl lO lO 

no oi co oi oi 


OI b- 05 O rp OI CO lO rH 


o ^ a io 

lO CO CO Ci rH 


H t- Tji o H Ol O X LO O 01 W H H t- 


Ol o o o c 

b* © CO b- CO 


00 

01 ^ 


LO 


HP 


CO 


50 

a 

3 

o 

P»H 

o 


PQ 


w 


M 

<1 


PQ 


PQ 


o 


Olnpl— Clt^GDrHCOOlrHrH 


Oi b* Hp CO hP 
lO 01 CO Ci rH 


Hp CO H CO QC 00 C Oi' CO H OI H hP GO 


| H OI b- lO GO 

co oi oi go oi 


1* 

-Q 


CO LO © OI rH CO t- OI OI -HH 


C 0 l 010 l'* 10 HCOO'tC 0 l-G 0 


DG 1C X X H H H l- 01 H 01 


b- 01 LO b- Oi 

GO tH CO b- tH 


OI OI LO t— b- 

Oi CC OI no OI 


CO CO CO O GO 

t- O CO t* H 


rHOlrHHrCOb-OlCiCOCinOb-rHCO 


HHHHGOGOOICOGCHOIOI • rH 


Oi O UO rH Oi 

oi i- oi lo oi 


Oi Hp t— b- b- 
GO GO CO hP tH 


OI 


OI 


COrHCOCiOlGOCOOiCDHPCi 


lO lO b- GO 110 
rH GO oi CO OI 


OI HP GO GO GO 


Hp OI rH 


HP OI •'+ GO lO 

O b- CO CO OI 


OI •OlO'plOGOOOOXlOH 


no lO HP rH lO 
rH GO oi hP Oi’ 


5 © HH 01 CO -P LO CO l- CO Ci O rH 01 CO ”P lO GO 


■r g 
J S O 

rr; O 4 ) 
OK 


KH 

'oTcO^io 5 . . 

' ' '-r w JO pH rH 


(1) Average number of sums in 15 seconds. (4) Number of seconds required to tbink •> combinations. 

(2) Number of seconds required to do 5 combinations. (5) Average deviation in number of combinations. 

(3) Number of seconds required to copy 5 answers, Table XVI, 1. 























































































































































SHOWING FOR A AND I> CLASS PUPILS OF THE FOURTH GRADE THE NUMBER OF SUMS IN THE VARIOUS GROUPS WRITTEN IN 10 SECONDS, 
THE NUMBER OF ANSWERS COPIED IN 5 SECONDS, AND GIVING THE TIME REQUIRED TO “THINK” TILE ANSWERS TO THE SAME GROUPS. 




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PQ 

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03 

5 


* 


fl 

























o> 


















































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T* 

























P-H 

5 






















/: 

2Q 

0) 
























TD 

rO 
























© tH Ol CO tH LO © b- X © © rH OI CO rH LO 


5? 


rH Ol CO rf LO © 


Note. — (1) Average number of answers in 5 seconds. (4) Standard deviation in number of combinations. 

(2) Number of seconds required to copy 5 answers. (5) I’. E. in number of combinations. 

(3) Average deviation in number of combinations. (0) F. E. in seconds to write 5 answers. 




















































































































































TABLE XVII —Part 2. Sums. 


85 





GO 


< 


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ft 


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rH ... 

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ft ft 00 00 b- • • • 

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rH Cl CO T ID ft l'- 00 


Note.— (1) Average number of sums in 10 seconds. (5) Average deviation in number of combinations. 

(2) Number of seconds required to do 5 combinations. (6) Standard deviation in number of combinations. 

(3) Number of seconds required to copy 5 answers. (7) P. E. in number of combinations. 

(4) Number of seconds required to “think” 5 combinations. (8) P. E. in seconds to do five combinations. 




















































































































































8G 





























































































































































































































































































































































































































































































































































































































































































































































































































































































































































































Discussion .—A study of Graph IX. demonstrates the growth in 
skill from the 3-B to the 4—A class. It also shows a rapid slowing 
up on the work as the groups contain larger numbers. From the 
standpoint of skill, we can, therefore, say that difficulty in addition 
increases as the magnitude of the numbers increase—that is, 
within the limits of the forty-five elementary facts. 

Tables XVI and XVII will sufficiently demonstrate the method 
pursued in the study of facility in the use of the combinations to 
justify the omission of the details of Tables XVIII and XIX ex¬ 
cept as to the averages, &c., which are given in detail below. The 
original tables of frequency will be found on file with other details 
of this investigation which space will not allow us to give here. 


88 


TABLE XVIII. 

SHOWING FOR A AND B CLASS THIRD GRADE PUPILS THE NUMBER OF PRODUCTS 
WRITTEN IN 10-25 SECONDS IN THE VARIOUS MULTIPLICATION GROUPS, THE 
NUMBER OF ANSWERS COPIED IN 5 SECONDS, AND GIVING THE TIME REQUIRED 
TO “THINK” THE PRODUCTS FOR THE SAME GROUPS. 

1. Answers. . 


Groups . 

( 1 ) 

( 2 ) 

(3) 

(4) 

(5) 

( 6 ) 

1 —A . ' 

9.04 

2.76 

1.6 

2.07 

0.15 

0.05 

B . 

7.73 

3.23 

1.5 

1.98 

0.19 

0.08 

2 —A .' 

9.73 

2.76 

1.6 




B . 

7.92 

3.16 

1 .61 




3—A . 

8.93 

2.8 

1.98 




B . 

7.23 

3.45 

1.61 




4—A . 

7.54 

3.31 

1.46 






(2.65) 





B . 

5.94 

4.21 

1.13 






(3.36) 





5—A . 

5.76 

4.37 

1.59 




B . 

5.55 

4.50 

1.00 




G—A . 

6.32 

3.96 

1 22 

1.87 

0.14— 

0 09_ 

B . 

4.8 

5.21 

0.86 

1.16 

0.11 + 

0.12 

7—A . 

5.98 

4.18 

1.05 




B . 

4.45 

5.62 

0.92 




8 —A . 

5.65 

4.42 

1.00 

1.45 

0 . 11 — 

0.085 

B . 

4.73 

5.28 

0.98 

1.52 

0.15— 

0.16 + 

9 — A . 

6.01 

4 . 16 

1.3 




B . 

4 . 45 

5.62 

1.06 




10 — A . 

5 . 56 

4.5 

1.44 

2.15 

0.18 + 

0.15 — 

B . 

O Q 

O . O 

6.58 

0.7 

0.91 

0.09— 

0.15 

11 —A . 

5.51 

4 . 54 

1.00 

1.29 

0 . 10 — 

0 08 

B . 

4.57 

5.47 

0.72 

1.07 

0 . 10 + 

0.12 + 

12 —A . 

6.07 

4.12 

1.46 




B . 

4.43 

5.64 

0.74 




13 — A . 

5.88 

4.25 

1.2 

1.65 

0.12 + 

0 09 — 

B . 

4.37 

5.72 

0.96 

1.26 

0.12 + 

| 0.16— 

14— A . 

5.63 

4.44 

1.07 

1.59 

0 . 12 — 

0.095 

B .. 

4.39 

5.72 

0.78 

1.07 

i o.io+ 

1 0.13 + 





























































89 


TABLE XVIII.— Continued. 

SHOWING FOR A AND B CLASS THIRD GRADE PUPILS THE NUMBER OF PRODUCTS 
WRITTEN IN 10-25 SECONDS IN THE VARIOUS MULTIPLICATION GROUPS, THE 
NUMBER OF ANSWERS COPIED IN 5 SECONDS, AND GIVING THE TIME REQUIRED 
TO “THINK” THE PRODUCTS FOR THE SAME GROUPS. 

1. Answers. 



Note. — (1) Average number of answers copied in 5 seconds. 

(2) Average number of seconds required to copy 5 answers. 

(3) Average deviation from the average number of answers. 

(4) Standard deviation. 

(5) P. E. in number of answers copied. 

(6) P. E. in seconds required to copy 5 answers of each group. 

(The numbers in parenthesis for groups 4 and 1G indicate the time required to 
copy the four answers constituting the group.) 





















90 


TABLE XIX. 


PRODUCTS. 


Groups .. . 

(1) 

(2)'- 

(3) 

(4) 

| (5) 

(6) 

(7) 

(8) 

1—A* . . . 

10.72 

4.63 

2.76 

1.9 

2.84 

3.33 

0.25 

0.11— 

P» .... 

9.47 

7.92 

3.23 

4.7 

3,4 

3.97 

0.38 + 

0.31— 

o_ a * 

11.57 

4 32 

2 76 

1.6 

2.4 




B . . . . 

11 .61 

6.49 

3.16 

3.3 

2.45 




3—A _ 

8.21 

6.09 

2.8 

3.3 

2.04 




B .... 

7.00 

10.71 

3.45 

7.3 

2.35 




4—A _ 

7.51 

6.66 

3 31 

3.35 

1.88 






(5.3) 

(2.6) 

(2.7) 





B .... 

7.25 

10.34 

4.21 

6.1 

2.3 






(8.3) 

(3.7) 

(4.9) 





5—A _ 

7.25 

6.89 

4.36 

2.5 

1.67 




B .... 

6.79 

11.05 

4.5 

6.55 

2.65 




6—A _ 

5.96 

8.39 

3.96 

4.4 

2.01 

co 

CD 

0.20 

0.28— 

B . ... 

5.61 

13.37 

5.21 

8.2 

2.27 

3.1 

0.28 + 

0.68 

7—A _ 

5.82 

8.59 

4.18 

4.4 

1.74 




B . . . . 

6.29 

11 .92 

5.62 

6.3 

2.02 




8—A .... 

5.27 

9.49 

4.42 

5.1 

1.85 

2.48 

0.19— 

0.34— 

B . . . . 

4.99 

15.03 

5.28 

9.75 

2.31 

2.75 

0.265 

0.8— 

9—A _ 

6.56 

7.64 

4.16 

3.5 

1 .98 




P> .... 

6.16 

12.17 

5.62 

6.55 

2.151. 



10—A _ 

8.19 

9.16 

4.5 

4.7 

1.88 

2.3 

0.17 + 

0.19 + 

Bt ... 

6.35 

14.97 

6.58 

8.4 

2.96 

3.31 

0.32— 

0.47 + 

11—A _ 

7.49 

10.01 

4.54 

5.5 

2.12 

2.65 

0.2— 

0.27— 

Bf ... 

7.18 

17.41 

5.47 

11.9 

3.33 

4.19 

0.39— 

0.94 + 

12—A _ 

7.4 

10.14 

4.12 

6.0 

1.93 




Bt . . . 

7.45 

16.79 

5.64 

11.15 

2.51 




13—A .... 

7.56 

9.93 

4.25 

5.7 

1.83 

2.26 

0.17— 

0.22 + 

Bt ... 

7.8 

16.03 

5.72 

11.3 

3.41 

4.08 

0.39 + 

0.81— 

14— A _ 

6.9 

10.87 

4.44 

6.4 

2.11 

2.65 

0.2— 

0.31 + 

Bt ... 

8.2 

15.24 

5.72 

10.5 

3.43 

3.78 

0.36 + 

0.67 + 

15—A .... 

6.34 

11.83 

4.75 

7.1 

1.76 




Bt ... 

8. OS 

15.47 

6.01 

9.5 

2.80 












































91 


TABLE XIX.— Continued. 

PRODUCTS. 


Groups ... 

(1) 

(2) 

(3) 

(4) 

(5) 

(6) 

(7) 

(8) 

16—A _ 

5.86 

12.80 

4.94 

7.9 

2.15 

2.6— 

0.195 

0.43— 



(10.2) 

(3.9) 

(6.3) 





Bt ... 

6.7 

18.66 

6.19 

12.5 

2.61 

3.06 

0.29 + 

0.9S + 



(11.9) 

(4.9) 

(10.0) 






Note. —No mark after class letter indicates that 15 seconds were given for each 
group. The * after the class letter shows that 10 seconds were allotted. The t 
after the class letter indicates that 25 seconds were given to the corresponding 
groups. 

(1) Average number of products written. 

(2) Number of seconds required to write 5 products. 

(3) Number of seconds required to copy 5 answers. (From Table XVIII, 1.) 

(4) Number of seconds required to “think” 5 combinations. 

(5) Average deviation from average (1). 

(6) Standard deviation. 

(7) P. E. in number of combinations written in time allowed. 

(8) P. E. in number of seconds required to write 5 combinations. 

(The numbers in parenthesis for groups 4 and 16 indicate the time required to 
write the four products in the group.) 


















92 


TABLE XX. 

SHOWING FOR A AND B CLASS PUPILS OF THE FOURTH GRADE THE AVERAGE NUM¬ 
BER OF ANSWERS COPIED IN 5 SECONDS, THE NUMBER OF SECONDS REQUIRED 
TO COPY FIVE ANSWERS, ETC. 


Answers. 


1 

Groups .| 

i 

(1) 

1 

(2) | 

(3) 

(4) 

(5) 

(6) 

1—A . 

10.88 

2.3 

1.78 

2.1 

0.142 

0.03 

B . 

8.94 

2.8 

1.22 

1.7 

0.115 

0.036 

2—A . 

10.45 

2 4 

1.51 




B . 

8.71 

2.87 

0.84 




3—A . 

9.38 

2.67 

1.58 




B . 

7.98 

3.13 

0.96 




4—A . 

7.6 

3.19 

1.27 





(2.63) 




B . 

6.87 

3.64 

0.72 






(2.9) 





5—A . 

7.03 

3.55 

1.39 




B . 

5.83 

4.29 

0.70 




6—A . 

6.66 

3.75 

1.13 

1.44 

0.097 

0.054 

B . 

6.00 

4.17 

0.66 

0.94 

0.063 

0.044 

7—A . 

5.92 

4.22 

1.03 




B . 

5.23 

4.78 

0.61 




8—A . 

5.93 

4.22 

0.89 

1.1 

0.074 

0.052 

B . 

5.20 

4.81 

1 

0.78 

0.95 

0.064 

0.059 

9—A . 

5.80 

4.31 

1.00 




B . 

5.07 

4.93 

0.68 




10—A . 

5.47 

4.57 

1.19 

1.46 

0.098 

0.1 

B . 

4.98 

5.02 

0.61 

0.88 

0.059 

0.06 

11—A . 

6.05 

4.13 

1.09 

1.39 

0.094 

0.064 

B . 

5.08 

4.92 

0.56 

0.81 

0.054 

0.052 

12—A . 

6.20 

4.03 

1.09 




B . 

5.01 

5.00 

0.58 




13—A . 

6.21 

4.02 

1.00 

1.26 

0.085 

0.055 

B . 

5.05 

4.95 

0.69 

0.88 

0.059 

0.057 

14—A . 

5.96 

4.19 

0.93 

1.25 

0.084 

0.059 

B . 

4.89 

5.11 

0.62 

0.81 

0.054 

0.056 


































































93 


TABLE XX.—Continued. 

SHOWING FOR A AND B CLASS PUPILS OF THE FOURTH GRADE THE AVERAGE NUM¬ 
BER OF ANSWERS COPIED IN 5 SECONDS, THE NUMBER OF SECONDS REQUIRED 
TO COPY FIVE ANSWERS, ETC. 


Answers. 


Groups . 

(1) 

(2) 

1 

(3) 

(4) 

(5) 

(6) 

15—A . 

5.67 

4.41 

1.08 




B . 

4.72 

5.3 

0.67 




16—A . 

5.23 

4.78 

1.00 

1.15 

0.078 

0.071 



(3.8) 





B . 

4.23 

5.91 

0.62 

0.80 

0.054 

0.075 



(4.7) 






Note.— For explanation, see note to Table XVIII, 1. 























94 


TABLE XXI. 

SHOWING FOR A AND B CLASS PUPILS OF THE FOURTH GRADE THE NUMBER OF 
PRODUCTS OF THE VARIOUS MULTIPLICATION GROUPS WRITTEN IN 10 SECONDS, 
ALSO THE NUMBER OF SECONDS REQUIRED TO ‘‘THINK” THE FIVE (OR FOUR) 
COMBINATIONS OF EACH GROUP. 


Groups. 

(1) • 

(2) 

| (3) 

(4) 

(5) 

(6) 

(7) 

| (8) 

1—A . 

10.66 

4.69 

2.3 

2.4 

2.8 

3.07 

0.207 

0.092 

B . 

7.94 

6.3 

2.8 

3.5 

2.2 

2.63 

0.177 

0.114 

2—A . 

11.16 

4.48 

2.4 

2.1 

2.56 




B . 

7.81 

6.4 

2.S7 

3.5 

2.75 




3—A . 

10.16 

4.92 

2.67 

2.3 

2.1 




B . 

7.35 

6.8 

3.13 

3.7 

l.S 




4—A . 

9.75 

5.13 

3.19 

1.9 

1.6 






(4.1) 

(2.6) 

(1.5) 





B . 

6.65 

7.5 

3.64 

3.9 

1.5 






(6.0) 

(2-9) 

(3.1) 





5—A . 

8.18 

6.11 

3.55 

2.6 

1.7 




B . 

5.46 

9.16 

4.29 

4.9 

1.4 




G—A . 

7.89 

6.34 

3.75 

2.6 

1.3 

1.55 

0.105 

0.084 

B . 

4.S6 

10.4 

4.17 

6.2 

1.4 

1.7 

0.115 

0.24 

7—A . 

6.99 

7.2 

4.22 

3.0 

1.5 




B . 

4.07 

12.28 

4.78 

7.5 

1.44 




8—A . 

6.58 

7.6 

4.22 

3.4 

1.9 

2.074 

0.139 

0.16 

B . 

3.34 

14.97 

4.81 

10.2 

1.5 

1.S4 

0.124 

0.56 

9—A . 

7.77 

6.43 

4.31 

2.1 

1.4 




B . 

3.92 

13.01 

4.93 

8.1 

1.3 




10—A . 

7.12 

7.02 

4.57 

2.4 

1.7 

2.06 

0.139 

0.137 

B . 

3.88 

12. SS 

5.02 

7.9 

1.2 

1.46 

0.098 

0.325 

11—A . 

5.84 

8.56 

4.13 

4.4 

1.7 

2.16 

0.146 

0.215 

B . 

2.2 

22.73 

4.92 

17.8 

1.2 

1.37 

0.092 

0.95 

12—A . 

6.46 

7.74 

5.03 

2.7 

1.7 




B . 

2.66 

18.8 

5.0 

13.8 

1.3 




13—A . 

5.91 

8.46 

4.02 

4.4 

1.8 

2.2 

0.148 

0.212 

B . 

3.26 

15.34 

4.95 

10.4 

1.6 

1.7 

, 0.114 

0.536 

14—A . 

6.19 

8.3 

4.19 

3.9 

1.7 

2.02 

0.148 

0.192 

B . 

2.90 

17.24 

5.11 

12.1 

0.9 

1.45 

0.098 

0.571 


























































95 


TABLE XXL—Continued. 

SHOWING FOR A AND B CLASS PUPILS OF THE FOURTH GRADE THE NUMBER OF 
PRODUCTS OF THE VARIOUS MULTIPLICATION GROUPS WRITTEN IN 10 SECONDS, 
ALSO THE NUMBER OF SECONDS REQUIRED TO “THINK” THE FIVE (OR FOUR) 
COMBINATIONS OF EACH GROUP. 


Groups . 

(1) 

(2) 

(3) 

(4) 

(5) 

(6) 

(7) 

(8) 

15—A . 

4.97 

10.06 

4.41 

5.0 

1.4 




B . 

2.3S 

21.0 

5.3 

15.8 

1.3 




16—A . 

4.82 

10.37 

4.78 

5.6 

1.7 

2.0 

0.135 

0.29 



(8.3) 

(3.8) 

(4.5) 





B . 

1.53 

32.68 

5.91 

26.8 

0.6 

1.037 

0.060 

0.148 



(26.0) 

(4.7) 

(21.4) 






Note.— For explanation, see note to Table XIX. 


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97 


Discussion. —In giving these multiplication tests the sixteen 
groups of facts were divided into two parts, the first part consist¬ 
ing of the first nine groups in their numerical order, and the 
second part of the last seven groups. These groups were given 
first in their regular order, then the second part was given first, 
and finally in the third test the order of the groups was reversed. 
This plan had for its object the equalizing of the effects of fatigue, 
distributing it as nearly as possible throughout the whole series. 

A study of Graph X shows a tendency upward throughout the 
“curves” in somewhat the same manner as the “curve” of errors 
in Grapji VI. The most remarkable thing about this figure is the 
irregularity of the “curve” for the 4—B class. In the natural 
course of events it should fall between the “curve” for 3—A and 
that for 4-A as in Graph IX. This curve is doubtless due to the 
fact at the time these tests were given the 4-B class had been 
engaged for some time in studying the addition and subtraction 
of simple fractions. The test caught the 3—B class drilling on 
the latter part of the tables, hence the increase of skill noted 
in that class from the 11th group on to the end. 

The two values given in groups 4 and 16 are due to calcula¬ 
tion on the basis of both 4 and 5 combinations, the 5 combina¬ 
tion value being the higher. 

An analysis of the four “curves” shows that groups 3, 6, 8, 
11, and 16 present a marked degree of difficulty from the stand¬ 
point of skill, as compared with the groups adjacent to them. 
Groups 2, 4, 7, 9, and 12 are correspondingly easy. The fact that 
group 1 shows higher difficulty than group 2 is doubtless due to 
the errors of process made in 1 X 1? which were in these tests 
counted as errors. Arranged then according to their relative dif¬ 
ficulty, beginning with the easiest, the groups would take the 
following order: 2, 1, 4, 3, 5, 9, 10, 7, 6, 8, 12, 13, 14, 15, 11, 16. 

A comparison of Graph X with the per cent, of errors part of 
Graph VI shows that skill in the manipulation of the combinations 
as represented by time is in inverse ratio to the number of errors 
made, the high points of each “curve” being in practically the 
same relative positions. The like is also observed in the addi¬ 
tion by a comparison of Graphs V and IX. In other words, the 


groups showing the greatest skill in handling are the groups in 
which the fewest errors appear. Rapidity in this case goes with 
fewness of errors. 


XVIII. CONCLUSIONS. 

1. In the process of learning each group of combinations power 
is developed whereby subsequent groups are more easily mastered. 

2. Girls learn the addition combinations with less difficulty 
than boys by about 10 %, and make 10% fewer errors through¬ 
out the process. 

3. Girls learn the multiplication combinations with about 25% 
less difficulty than boys, and make about 13% fewer errors in 
the process of mastery, but boys retain 2.7% better than girls 
when the whole number of products is considered, excelling es¬ 
pecially in the larger ones, while girls excel slightly in retain¬ 
ing the smaller products. 

4. In the process of teaching the addition facts fewer errors 
appear in the double numbers even when equal opportunity for 
making errors is provided, the fewest appearing in 5 + 5 and 
the greatest number in 8 + 8. 

5. In teaching multiplication the fewest errors appear in the 
10’s and ll’s and the next fewest in the doubles after 3X3. 

6. Combinations in the latter part of the series in both addi¬ 
tion and multiplication are more readily learned, but they are 
also more easily forgotten. 

7. From the standpoint of skill in manipulation, difficulty in 
addition increases with the magnitude of the numbers. 

8. The groups showing least skill for their manipulation are 
the same as those showing the greatest number of errors. This 
shows close correlation between speed and accuracy, greater speed 
going along with greater accuracy. 

9. The story of relative difficulty for addition facts is told in 
detail by Graphs V and VII. 

10. The story of relative difficulty for multiplication combina¬ 
tions is told in detail by Graphs VI and VIII. 


99 


XIX. SUGGESTIONS GROWING OUT OF THE 
INVESTIGATION. 

1. The combinations in both addition and multiplication should 
be taught in smaller groups than the ordinary tables. Five com¬ 
binations per group is suggested as best, giving as it does an 
average of one new combination for each school day of the week. 

2. In a group of five, or any other number of facts to be 
learned, it should be remembered that the first and the last posi¬ 
tion are points of vantage, and that the hardest combinations of 
a given group should be placed at these two points, and the 
easier ones in the middle. 

3. Subtraction should be taught simultaneously with addition, 
and as an additive process. 

4. Division should be taught simultaneously with multiplica¬ 
tion. 

5. It is not economical to begin drills in addition facts before 
the beginning of the second half of the first school year. 

6. The multiplication combinations may profitably be begun 
in the second half of the second school year. They may all be 
taught in one term, if that term falls in the second half of the 
year, but it is not advisable to attempt beyond 10 X 10 in this 
term. 

7. In both addition and multiplication, combinations should 
be taught in both direct and reverse forms simultaneously; as, 
2 + 3 = 5, 3 + 2 = 5; or 6 X I =42, 7 X 6 = 42. 

8. No scheme of association, no number plays or games which 
have not in them as their predominant factor the idea and essence 
of drill, can take the place of systematic and persistent repeti¬ 
tion as a means of making permanent and automatic the correct 
reactions to the combinations in addition and multiplication. 

Education has too long been philosophical. Every phase of 
the subjects taught in the schools is fairly bristling with un¬ 
solved problems—problems that are “settled” to-day in a dog¬ 
matic way and unsettled to-morrow in a dogmatic way, the sta¬ 
bility of the decisions depending, upon the position and the per¬ 
sistence and the eloquence of the individual who renders the 


100 


dictum. While the day of dictum is not over, the dawn of a new 
order of things in educational procedure has already begun. 
The only way in which pedagogy may write its name perma¬ 
nently among the sciences is by basing its conclusions upon the 
incontrovertible evidence of scientific investigation. This task 
is no easy one. It is too big for the individual investigator. He 
can at best only point the way. Such a study as this may posit 
conclusions only tentatively. There should be in every large 
system of education a department for the investigation of the 
problems of the school room. This department might hope to 
make its investigations sufficiently extensive to render its con¬ 
clusions universally valid. Every large industrial establishment 
maintains such a department. It should be possible in the work 
of education. 

BIBLIOGRAPHY AND REFERENCES. 

1. Branford, Benchara, A Study of Mathematical Education, 
Ch. V. 

2. Doring, Max, Zur Psychologie des Kleinen Einmalein, Zeit. 
f. Pad. Psy., Vol. XIII, Xo. 3, pp. 163-171. 

3. Gildemeister, Theda, The Multiplication Tables, pp. 4-31. 

4. Idem, p. 4. 

5. Griggs, A. O., The Pedagogy of Mathematics, Pedagogical 
Seminary, Vol. XIX, Xo. 3, p. 360. 

6. Harris, W. T., Rosenkranz’s Philosophy of Education, Xote 
p. 91. 

7. Idem, p. 93. 

8. Idem, p. 251. 

9. Henmon, V. A. C., The Relation Between the Mode of 
Presentation and Retention, Psy. Rev., Vol. 19, pp. 79-96. 

10. Holloway, H. V., The Effect on Retention of a Varying 
Humber of Initial Repetitions of a Memory Gem, MS., Depart¬ 
ment of Education, University of Pennsylvania. 

11. James, Wm., Psychology (Briefer Course), p. 145 and 
p. 298. 

12. Kuhlman, F., the Present Status of Memory Investiga¬ 
tion, Psy. Bui. (1908), Vol. 5, pp. 285-293, also On the Analysis 


101 


of the Auditory Memory Consciousness, Am. Jour, of Psy. 
(1909), Vol. 20, pp. 194-218. 

13. Leipzig Teachers’ Association, Report of the Arithmetic 
Committee of the, Second Introduction (1908). 

14. Los Angeles, Cal., Course of Study (1910-1911), pp. 
155-156. 

15. McDougle, Ernest C., A Contribution to the Pedagogy of 
Arithmetic, Ped. Sem., Vol. XXI, Xo. 2, June, 1914, 161-218. 

16. McLellan and Dewey, The Psychology of Xumber, p. 61. 

17. McMurry, Charles A., Special Method in Arithmetic, pp. 
53-54. 

18. Idem, pp. 54-55. 

19. Idem, pp. 54 et seq. 

20. Meumann, E., Okonomie und Technik des Gedachtnisses, 
Zeit. f. pad. Psy. (1906), Vol. 8, pp. 329-343. 

21. Xew Jersey, State Monograph on the Teaching of Arith¬ 
metic (1912), pp. 40 and 44. 

22. Phelps, C. L., A Study of Errors in Tests of Adding Abil¬ 
ity, Elementary School Teacher, Vol. 14, Xo. 1, Sept. 1913, pp. 
29-39. 

23. Phillips, D. E., Xumber and Its Application, Pedagogi¬ 
cal Seminary, Vol. 5, p. 254. 

24. Idem, p. 233. 

25. Pohlman, A., Experimentelle Beitrage, Zur Lehre vom 
Gedachtnis, Gerdes, Berlin (1906), p. 191. 

26. Quigley, E. M., The Common Sense Method of Primary 
Xumber Work (1901). 

27. Ranschburg, P., Zur physiologischen und pathologischen 
Psychologie der elementaren Rechenarten, Zeit. f. exp. Pad., Bd. 
7, 135-162, 1908; Bd. 9, 1909, 251-263. 

28. Rigler, Frank, Course of Study in Arithmetic for Portland, 
Oregon, MS., p. 28. 

29. Idem. 

30. Segal, J., Ueber den Reproduktion stypus und Repro- 
duzierem von Vorstellungen, Arch, fiir die ges. Psy. (1908), Vol. 
12, pp. 124-236. 

31. Smith, D. E., The Teaching of Arithmetic, Teachers Col¬ 
lege Record, Vol. X, Xo. 1, p. 54 (1909). 


102 


32. Idem, p. 65. 

33. Idem, p. 67. 

34. Speer, Method in Arithmetic. 

35. Speer, Theory of Arithmetic, Primary Book for Teachers, 
p. 34. 

36. Stone, C. W., Problems in the Scientific Study of the 
Teaching of Arithmetic, Journal of Educational Psychology, Vol. 
IV, Xo. 1, pp. 1-16 (1913). 

37. Thomdyke, E. L., Memory for Paired Associates, Psy. Rev. 
(1908), Vol. XV, pp. 122-137. 

38. Thorndyke, E. L., On the Function of Visual Images, Jour, 
of Phil. Psy. (1907), Vol. 4, pp. 324—327, vide also (33). 

39. Thorndyke, E. L., Mental and Social Measurements, Ch. 
VI, The Science Press (1904), p. 210, also p. 139 et seq. 

40. Von Sybel, A., Ueber das zusammenwirken verschiedener 
Sinnesgebiete bei Gedachtnisleistungen, Zeit. fur Psy. und Phys. 
des Sinnes, Vol. 53, Abt. 1, pp. 257-360. 

41. Weber, E., ITntersuchungen zur Psychologie des Gedacht- 
nisses mit Einleitung liber die bisherigen Versuehe zur experi- 
mentellen Forschung des Gedachtnisses, Zeit. fur exp. Pad. 
(1909), Vol. 8, pp. 1-81. 

42. White, E. E., Elements of Pedagogy, p. 296. 

43. Wood, Elizabeth L., Methods of Presentation and Learn¬ 
ing, Ped. Sem., Vol. 19, pp. 250-279. 

44. Yocum, A. Duncan, A First Step in Inductive Research 
into the Most Effective Methods of Teaching Mathematics, School 
Science and Mathematics, Vol. XIII, Xo. 3, p. 205 (1913). 

45. Yocum, A. Duncan, An Inquiry into the Teaching of Addi¬ 
tion and Subtraction, p. 40. 

46. Idem, pp. 35-36. 

47. Yocum, A. Duncan, Lectures on the Institutes of Pedagogy. 



















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